Sync catch-up sessions list server-side

### Problem I have

I love using the Catch-Up feature, and it brings me a lot of zen. However, when I switch from Phanpy on my phone to Phanpy on my computer and vice-versa throughout the day, I lose track of when I last did a catch-up, which leaves me feeling less in control of my time and attention and makes it harder to figure out where I was reading last and what I might have missed.

### Solution I'd like

I would like the list of previous catch-up windows to be sync'd through the private note on my own profile.

### Alternatives considered

Can't think of any

### Other

_No response_

No files, tests, or entry points are named. Start by locating the Catch-Up session list and the handling of private notes on the user's own profile, then determine how the list can be persisted and read across devices. Done means previous catch-up windows are synchronized when switching between phone and computer.

Written by the indexing model from the issue text.