[Feature Request] chat in Incognito Mode

## Describe the solution you'd like

don’t store any inputs or outputs from the model on the servers when you use "Incognito Mode"
When a user refreshes their page - then all previous context inputs and outputs are lost when you use "Incognito Mode"

## Why the solution needed

Users want mode where nothing saved when they have incognito mode

## Implementation feasibility

Are you willing to discuss the solution with us, decide on the approach, and assist with the implementation?

- [V] Yes
- [ ] No

No files, tests, or entry points are named in the issue. Begin by tracing where chat inputs and outputs are stored and how conversation context survives a page refresh, then define the client and server boundaries for Incognito Mode. Done means Incognito sessions store nothing on servers and lose all prior context after refresh, with that behavior verified.

Written by the indexing model from the issue text.