The thread has been updated with the latest PKHeX version, which is now based on .NET 10. Fortunately, there are no major regressions, unlike what happened during the transition to .NET 9. Just manually install the latest .NET Desktop Runtime 10 and you should be good to go (it's not available on Winetricks yet).
I’m not sure whether this is an Arch-specific issue, but the latest Wine release (10.20) prevents the program from starting. I’m currently using version 10.19.