Synchronize messages across clients (including WebUI)

We want to enable workflows that span different interfaces. Therefore, we want the messages that users interchanged with Duo Chat in one interface also show up in another interface. As an example, one might prepare an issue in WebUI using the chat and then continue to iterate in VS Code with a code change.

Status before this is implemented

Syncing already happens in the GitLab web app:

• in multiple tabs
• or when I reopen the chat
• or when a new message is sent in the IDE it gets sync'ed to the chat in the web app.

However, it doesn't work in the IDEs. We always start new. And we do not see changes from other clients coming to the IDE.

Proposal

Fetch all stored messages from API in the IDEs just like we do on the Web. It should happen every time we open chat in IDE.