Run TabFern in background even when not open (or option to minimise to tray)

One of the great features of TabsOutline is that I can open TO, open a tab group, then close TO. Meanwhile, any changes that I make to the opened tab group are always reflected in that group in TO. This is not the case with TabFern. I open TabFern, open my group, close TabFern but then any changes I make to that group are NOT reflected int eh saved group on TabFern next time I open it.

I can only assume that with TO, it is running constantly in the background whenever Chrome is running regardless of whether the TO tree window is open? So can TabFern mimic this? If that's not possible, then the next best thing would be add an option where closing the TabFern tree window with the "x" at top right of window does not close TabFern but minimises it to the tray. But to be totally honest I'd much prefer the "running at all times" version as per TO as it's totally foolproof and does not rely at all on me remembering to re-save the group in TabFern before I close that group.

The other irritating thing about the TabFern current behaviour is that if I've opened a group, closed the TF tree window, then want to update that group with moved/additional tabs, I re-open the TF tree window and that same group comes up as "unsaved" so I end up with two groups, the original one (which I then have to delete) and then this modified one. That's not helpful.

Max

The issue names no files, tests, or entry points. Reproduce the saved-group behavior by closing and reopening the TabFern tree window, then trace the extension lifecycle and decide which requested behavior is in scope; done means changes to an opened group persist without duplicate unsaved groups after reopening.

Written by the indexing model from the issue text.