Fix agent processing stopping when switching tabs within app

[Al629176]

Summary

Fix chat processing so agent thinking/loading continues even when user switches tabs inside the app (e.g., Chat -> Skills/Settings).

Problem

When a user sends a message on Chat and navigates to another tab, in-progress agent processing appears to pause or stop. Returning to Chat shows stuck/non-completing state.

Expected Behavior

• Agent processing should continue in background after tab switch.
• In-flight responses should complete regardless of active tab.
• Returning to Chat should show the completed or current in-progress state.

Scope

• Reproduce tab-switch interruption flow across Chat, Skills, Settings.
• Identify lifecycle/state coupling between chat processing and visible route.
• Decouple agent run state from chat screen mount/unmount behavior.
• Ensure sockets/tasks/subscriptions remain active across navigation.
• Add recovery/resync when Chat screen remounts.

Acceptance Criteria

• Message processing does not stop on internal tab switches.
• Agent responses complete successfully while user is on other tabs.
• Chat view rehydrates correct loading/completed state on return.
• Regression tests cover route change during in-flight responses.