[FEATURE] Let an existing Cowork session be assigned/moved to a project (API silently ignores chat_project_id on PUT)
A chat conversation can be moved between projects: PUT /api/organizations/{org}/chat_conversations/{uuid} with {"project_uuid": …} returns 202 and the change reads back.
There is no equivalent for a Cowork/Code session. PUT /v1/code/sessions/{cse_id} accepts chat_project_id, chat_project_uuid and project_uuid and returns 200 in every case while the field stays null — a silent no-op, which is worse than a 400. POST /v1/code/sessions/{id}/project and /move are 404, and PATCH|PUT /api/organizations/{org}/cowork/sessions/{id} and .../code/sessions/{id} are 404.
Result: a session that was started outside a project (including trigger-spawned relay runs) can never be filed, so an account that organises everything by project keeps a permanent unfileable remainder — 15 sessions on this account.
Asks, in order of preference:
- Allow the project to be set on an existing session (API + a "Move to project" control in the UI, matching chats).
- Failing that, reject an unsupported field with 400 instead of returning 200 and dropping it.
Related: #80061 (project assignment cosmetic), #86611 (project grouping).