I tried installing from macOS 11 with the installer script and it said macOS 12 was required. I tried from 12 and it said 13 was required. I tried from 13 and it worked and I upgraded to F41.
Then I downgraded to macOS 11. If I choose the KDE partition (macOS 13.5 stub) in startup options the Apple logo flashes three times - once for about two seconds, once for about half a second and once for about thirteen seconds - before loading macOS 11 login. If I select the macOS 13.5 stub (KDE partition) in Startup Disk in System Preferences I end up in an infinite boot loop.
It’s possible to dual-boot macOS 11 just fine, but we do not support wiping the macOS that you used to install Asahi Linux, and by “downgrading” presumably that’s what you did (I assume the “downgrade” was a wipe and reinstall since I’m pretty sure there’s no other way to downgrade macOS). That means you now have lost your machine owner credentials tied to the Asahi Linux partition, so you can’t fix the boot policy (which is probably what got screwed up somewhere along the way).
Furthermore, macOS 11 is too buggy and old to even install Asahi at this point for several reasons, nor can it manage macOS 12+ boot policies correctly, since it lacks the upwards-compatibility mechanism that Apple introduced in macOS 12, or even a working bless command for boot policy creation. So we couldn’t support working with this downgrade scenario even if we wanted to.
In other words, you need multiple macOS installs: macOS 13.5 or newer as the blessed version to install and maintain Asahi, and whatever older version(s) you want. If you ever run into Asahi bootability problems (not unlikely with a macOS 11 install, since it’s quite simply broken for this stuff), you will have to boot into 13.5+ and run a repair from there.
I would in general discourage usage of macOS older than 12.3 on any Apple Silicon machine except as the only OS. Apple didn’t have the multi-OS stuff figured out properly at launch at all. You’re likely to run into strange bugs and problems.