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I think Pachli on Android has this feature: https://play.google.com/store/apps/details?id=app.pachli
As for iOS, I wouldn't know.
https://docs.google.com/spreadsheets/u/0/d/1De5KRwqMIdwEryfoeBLARgxF7QgKkeOQBCilKuIdAXE/htmlview
Not sure if up to date but possibly useful. Comparison table for apps.
Fedilab, not the most pretty app, but the most functional one and with decent customization.
But it doesn't fetch new posts on launch, or maybe i'm missing some option
Settings > Home Cache > Automatically fetch home messages
I don't think thats it, it only allows me to set a timer for how often it fetches home messages
That's the idea. You fetch them in the background, instead of waiting 30 seconds every time you open the app. You can also enable the same setting in the Timeline.
I just double-checked and if you have Automatically fetch missing messages toggled in Settings > Timelines it will update on opening the app.
This doesn't seem to work for me idk. But i'm now trying fedilab with that timed fetching thing, since there isn't anything better either