feat(web): list background tasks in the session header

The task registry has run every background bash, pwsh, pty-send, and
one-shot subagent since it landed, but only the model could read it: a
human at the Web client could not see that a build was running, tell a
finished task from a stuck one, or find its outcome anywhere but the
`run_in_background` tool card that printed an id and never updated.

Task state now reaches the browser as one whole-snapshot `session/tasks`
mux frame per session, pushed at every registry commit that changes what
that session can see. `TaskService` gains `onTasksChanged`, which is
owner-granular because owner-disposal removal is a change no per-task
record can express. The carrier reads the exact owner the listener hands
it, so a push stays correct while that scope tears down, and reads the
baseline through the non-resuming `ctx.agents.get` so listing never
revives a cold session. The client keeps a last-wins mirror on
`SessionListState`, and a new `dsh-client-ui-task` package renders it
beside the subagent catalog — rendering nothing at all until the session
has a task, so an ordinary conversation grows no new chrome.

Streamed per-task output and human-initiated cancellation are separate
phases; the note records why neither has to undo this channel, and why
no Web path may call the consuming `ctx.tasks.read()`.

packages/client/ui-tool/tests/chat-code-subcalls.spec.tsx

@@ -101,7 +101,7 @@ async function bench(snapshot: ConversationSnapshot) {
     ids: [SID],
     byId: { [SID]: { id: SID, title: 'S', displayTitle: 'S', running: false, blank: false, updatedAt: 1 } },
     current: SID,
-    phase: 'ready', subagentsByParent: {}, currentAddress: undefined,
+    phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
   })
   const scoped = { send: vi.fn(async () => {}), cancel: vi.fn(async () => {}) }
   const layout = { openDetails: vi.fn(), closeDetails: vi.fn() }

packages/client/ui-tool/tests/coverage-tails.spec.tsx

@@ -30,7 +30,7 @@ function listStore() {
     },
     current: undefined,
     phase: 'ready',
-    subagentsByParent: {},
+    subagentsByParent: {}, tasksBySession: {},
     currentAddress: undefined,
   })
 }

packages/client/ui-tool/tests/diff-card.spec.tsx

@@ -157,7 +157,7 @@ describe('FileMutationRow diff card', () => {
     byId: { [SID]: { id: SID, displayTitle: 'r', running: false, blank: false, updatedAt: 0, cwd: '/w/app' } },
     current: SID,
     phase: 'ready',
-    subagentsByParent: {},
+    subagentsByParent: {}, tasksBySession: {},
     currentAddress: undefined,
   })
 
@@ -309,13 +309,13 @@ describe('DetailsPanel diff Output section', () => {
     const chat = createChatStore().create()
     if (selection !== null) chat.actions.select(selection)
     const sessions = createSnapshotStore<SessionListState>(cwd === undefined
-      ? { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, currentAddress: undefined }
+      ? { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined }
       : {
         ids: [SID],
         byId: { [SID]: { id: SID, displayTitle: 'r', running: false, blank: false, updatedAt: 0, cwd } },
         current: SID,
         phase: 'ready',
-        subagentsByParent: {},
+        subagentsByParent: {}, tasksBySession: {},
         currentAddress: undefined,
       })
     const workspaces = createSnapshotStore<WorkspaceListState>({

packages/client/ui-tool/tests/read-card.spec.tsx

@@ -171,7 +171,7 @@ describe('ReadRow keyed toolview', () => {
     byId: { [SID]: { id: SID, displayTitle: 'r', running: false, blank: false, updatedAt: 0, cwd: '/w/app' } },
     current: SID,
     phase: 'ready',
-    subagentsByParent: {},
+    subagentsByParent: {}, tasksBySession: {},
     currentAddress: undefined,
   })
 
@@ -255,13 +255,13 @@ describe('DetailsPanel Output section (read)', () => {
     const chat = createChatStore().create()
     if (selection !== null) chat.actions.select(selection)
     const sessions = createSnapshotStore<SessionListState>(cwd === undefined
-      ? { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, currentAddress: undefined }
+      ? { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined }
       : {
         ids: [SID],
         byId: { [SID]: { id: SID, displayTitle: 'r', running: false, blank: false, updatedAt: 0, cwd } },
         current: SID,
         phase: 'ready',
-        subagentsByParent: {},
+        subagentsByParent: {}, tasksBySession: {},
         currentAddress: undefined,
       })
     const workspaces = createSnapshotStore<WorkspaceListState>({

packages/client/ui-tool/tests/search-card.spec.tsx

@@ -378,7 +378,7 @@ describe('DetailsPanel Output section (search)', () => {
     if (selection !== null) chat.actions.select(selection)
     const sessions = createSnapshotStore<SessionListState>({
       ids: [], byId: {}, current: undefined, phase: 'ready',
-      subagentsByParent: {}, currentAddress: undefined,
+      subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
     })
     const workspaces = createSnapshotStore<WorkspaceListState>({
       items: [], archivedSessionIds: [], state: 'idle', phase: 'ready', error: null,

packages/client/ui-tool/tests/terminal-card.spec.tsx

@@ -346,7 +346,7 @@ describe('BashRow terminal card', () => {
     byId: { [SID]: { id: SID, displayTitle: 'r', running: false, blank: false, updatedAt: 0 } },
     current: undefined,
     phase: 'ready',
-    subagentsByParent: {},
+    subagentsByParent: {}, tasksBySession: {},
     currentAddress: undefined,
   })
 
@@ -446,13 +446,13 @@ describe('DetailsPanel Output section', () => {
     const chat = createChatStore().create()
     if (selection !== null) chat.actions.select(selection)
     const sessions = createSnapshotStore<SessionListState>(cwd === undefined
-      ? { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, currentAddress: undefined }
+      ? { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined }
       : {
         ids: [SID],
         byId: { [SID]: { id: SID, displayTitle: 'r', running: false, blank: false, updatedAt: 0, cwd } },
         current: SID,
         phase: 'ready',
-        subagentsByParent: {},
+        subagentsByParent: {}, tasksBySession: {},
         currentAddress: undefined,
       })
     const workspaces = createSnapshotStore<WorkspaceListState>({
@@ -643,7 +643,7 @@ describe('DetailsPanel Output section', () => {
         useSessions={bindSnapshotSelector(createSnapshotStore<SessionListState>(
           {
             ids: [], byId: {}, current: undefined, phase: 'ready',
-            subagentsByParent: {}, currentAddress: undefined,
+            subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
           }))}
         useWorkspaces={bindSnapshotSelector(createSnapshotStore<WorkspaceListState>({
           items: [], archivedSessionIds: [], state: 'idle', phase: 'ready', error: null,

packages/client/ui-tool/tests/web-card.spec.tsx

@@ -208,7 +208,7 @@ describe('DetailsPanel web Output section', () => {
     if (selection !== null) chat.actions.select(selection)
     const sessions = createSnapshotStore<SessionListState>({
       ids: [], byId: {}, current: undefined, phase: 'ready',
-      subagentsByParent: {}, currentAddress: undefined,
+      subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
     })
     const workspaces = createSnapshotStore<WorkspaceListState>({
       items: [], archivedSessionIds: [], state: 'idle', phase: 'ready', error: null,