It is a limitation of a PWA. A native app (wrapper) will likely be coming though, and a custom Capacitor plugin could theoretically interface with the native API to switch out icons. Theoretically. No promises though :)
Quoting this comment by the dev
With all browsers?, here runs fine with chromium.
No browser as far as I know. I installed the PWA and haven’t touched anything else.
I’m on iOS exclusively with no plans to use a desktop interface
The main issue is Safari, bugs in webkit bog everything down. And iOS is stuck with it.