Performance: Matrix client should only sync information relevant to the current page

[Stvad]

I noticed that the client seems to be doing sync for all the messages, and as a consequence it takes a while to load the highlights.
It should be configured to do sync with filter only on the relevant room by default.

[DanilaFe]

I've definitely had this in mind, but last time I looked into it, I remember running into trouble (the precise sort of trouble I do not currently recall). On the other hand, with the present architecture, there's a single Matrix client for all pages; this means it will at the very least need to switch filters whenever a new page is "opened".

[Stvad]

can probably route to a specific client based on source tab id/url 🤔 from event.

I'm also actually not convinced that running the client in the content script is such a bad idea 🤷‍♂️