Add saved conversations

We should be able to save conversations and go back to them, losing them on swap isn't ideal. With this I would also just assume we could directly grab more content from the page within this view and insert it, for exmple:

1. We summarize a video
2. We're taken to the summary of the video and the chat view
3. we leave and go to settings
4. we go back "home" however we should have an option for most recent conversation? (maybe) but at the very least saved conversations
5. We return to a saved conversation, but are on a new video
6. We get the transcription and ask for comparisons between the original and new one
7. ect.

This would greatly improve the usability of the extension.

No files, tests, or entry points are named. Start by mapping the extension's chat, settings, home, and video-summary flows, then define how saved and most-recent conversations relate to new video content. Done should include a clear, implemented way to save, return to, and use conversations across navigation.

Written by the indexing model from the issue text.