Chat interface loses context randomly

### The problem
Currently the chat interface is losing its context.

#### Tasks:
- [ ] Audit the problem, reproduce the bug
- [ ] Document a fix implementation plan
- [ ] Implement, verify and write up the solution

**Screenshots:**

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#### Relevant code for chat context persistence and loading

- Initialization of session and loading chat state:
- [`Co-Founder/founder-os/app/page.tsx` (HomePageContent, useEffect for session state)](https://github.com/CodeBunny09/co-founder/blob/f6678c5488654a853499551c48fad8c3f8a98eb4/Co-Founder/founder-os/app/page.tsx#L78-L112)
- Persisting chat state and session to storage:
- [`Co-Founder/founder-os/lib/conversationState.ts` (saveConversationState)](https://github.com/CodeBunny09/co-founder/blob/f6678c5488654a853499551c48fad8c3f8a98eb4/Co-Founder/founder-os/lib/conversationState.ts#L59-L98)
- API logic for loading session state:
- [`Co-Founder/founder-os/app/api/chat/state/route.ts` (GET handler)](https://github.com/CodeBunny09/co-founder/blob/f6678c5488654a853499551c48fad8c3f8a98eb4/Co-Founder/founder-os/app/api/chat/state/route.ts#L21-L47)
- Known issues, context and task linkage discussion:
- [`Co-Founder/BUILD_STATUS.md` (Known issues)](https://github.com/CodeBunny09/co-founder/blob/f6678c5488654a853499551c48fad8c3f8a98eb4/Co-Founder/BUILD_STATUS.md#L29-L35)
- Message sending and additional session management:
- [`Co-Founder/founder-os/app/page.tsx` (sendMessage, local state updates)](https://github.com/CodeBunny09/co-founder/blob/f6678c5488654a853499551c48fad8c3f8a98eb4/Co-Founder/founder-os/app/page.tsx#L146-L187)
- Execution page context handling:
- [`Co-Founder/founder-os/app/execution/page.tsx` (sendChat)](https://github.com/CodeBunny09/co-founder/blob/f6678c5488654a853499551c48fad8c3f8a98eb4/Co-Founder/founder-os/app/execution/page.tsx#L477-L513)

Start by reproducing the context loss, then read the session-loading useEffect and sendMessage logic in Co-Founder/founder-os/app/page.tsx. Trace saveConversationState in lib/conversationState.ts through the GET handler in app/api/chat/state/route.ts, and compare the execution page's sendChat flow. Done means the cause is identified, the fix is implemented and verified, and the solution is documented.

Written by the indexing model from the issue text.