Completed desktop task bars were only being dismissed locally, which let the
persisted task list resurface on refresh or bleed into the next user turn.
This wires the existing server-side reset path into the desktop stores and
session flow so a finished task cycle is summarized once, cleared locally,
and removed remotely before the next round starts.
Constraint: Existing task persistence already lives behind the server task-list API and must stay compatible with persisted JSON task files
Rejected: Only hide the completed task bar in UI state | left stale persisted tasks behind and reintroduced them on reload
Rejected: Clear desktop state without a task summary | dropped useful completion context from the chat transcript
Confidence: high
Scope-risk: moderate
Reversibility: clean
Directive: Keep desktop task dismissal and task-list persistence behavior aligned; do not reintroduce local-only clearing without covering reload and next-turn flows
Tested: `cd desktop && bun run lint`
Tested: `cd desktop && bun run test src/stores/cliTaskStore.test.ts src/stores/chatStore.test.ts src/components/chat/SessionTaskBar.test.tsx src/components/chat/ComputerUsePermissionModal.test.tsx`
Tested: `bun test src/server/__tests__/e2e/business-flow.test.ts --test-name-pattern "Task Lists API"`
Not-tested: Full `bun test src/server/__tests__/e2e/business-flow.test.ts` suite still has unrelated pre-existing failures in Models and Sessions sections
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