## Why
WebSocket handshakes did not reuse the HTTP cookie store or retain response cookies, so routing cookies such as `__oailb` were unavailable to subsequent connections.
## What changed
- Reuse the HTTP factory's ChatGPT cookies for secure WebSocket handshakes, preserving explicit `Cookie` headers and marking generated headers sensitive.
- Retain allowlisted infrastructure cookies from both successful and rejected upgrades. Keep configured cookies scoped to their factory and exclude account and session cookies from the shared store.
- Apply HTTPS cookie scope to `wss` requests, preserving host and path restrictions and excluding insecure `ws` requests and non-ChatGPT hosts.
## Testing
Add HTTP/WebSocket cookie-sharing coverage and local TLS handshake tests for routing-cookie reuse across connectors, rejected-upgrade refreshes, explicit header precedence, cookie scope, session-cookie exclusion, and deletion.
GitOrigin-RevId: 6bd6afe16e97cf9758ca7ba207a4e88c969497a4
12acac2a66