lightdm fails to launch i3wm

Pick a user: lightdm finds entries for i3 and i3-with-shmlog. If you go back and switch to another user, it won't find the entries anymore.
If you do manage to select i3 or i3-with-shmlog it will fail to launch either of them and fall back to mate.

The logs are not particularly helpful, the only file showing an explicit error being seat-0greeter.log:

  [+4.08s] DEBUG: session-list.vala:57: Adding session i3 (i3)
  [+4.08s] DEBUG: session-list.vala:57: Adding session i3-with-shmlog (i3 (with debug log))
  [+4.08s] DEBUG: session-list.vala:57: Adding session mate (MATE)
  [...]
  [+32.81s] DEBUG: arctica-greeter.vala:251: Session i3 is not available, using system default mate instead
  [+32.81s] DEBUG: Starting session mate

Have edited mate.desktop to launch an xterm, and have used that to launch i3. i3 by itself works fine.
Have installed jwm to see if this affected other window managers. jwm works fine.

Happens on an install upgraded from ubuntu mate 20.04.3 to 22.04.1 a few days ago.
Has happened to a friend on a clean install of 22.04 last summer, gave up after a couple hours trying to figure it out and assumed it was user error.
Has been reproduced inside a vm.