will only come up on second seat once [loginctl multiseat]

I've set up a second seat via loginctl --attach, resulting in the following udev rules:

TAG=="seat", ENV{ID_FOR_SEAT}=="drm-pci-0000_6e_00_0", ENV{ID_SEAT}="seat1"
TAG=="seat", ENV{ID_FOR_SEAT}=="input-pci-0000_6c_00_0-usb-0_11_1_0", ENV{ID_SEAT}="seat1"
TAG=="seat", ENV{ID_FOR_SEAT}=="sound-pci-0000_6e_00_1", ENV{ID_SEAT}="seat1"
TAG=="seat", ENV{ID_FOR_SEAT}=="usb-pci-0000_6e_00_3-usb-0_1", ENV{ID_SEAT}="seat1"

Because of #2033322, WaylandEnable=false is set in /etc/gdm3/custom.conf.

It works great--except it does so only once. I can login on seat1 just fine, except when I log out, the greeter doesn't come back up again, it just leaves me with a black screen.
Can be recovered from by restarting gdm (via ssh from another box or from seat0), but obviously that isn't ideal.

I've attached a pstree for the working case (no GUI logins, both greeters up) and the non-working post-logout one (no GUI logins, only seat0 hat a greeter up); please advise re. what logs to provide.

ProblemType: Bug
DistroRelease: Ubuntu 22.04
Package: gdm3 42.0-1ubuntu7.22.04.3
Uname: Linux 6.4.12-060412-generic x86_64
ApportVersion: 2.20.11-0ubuntu82.5
Architecture: amd64
CasperMD5CheckResult: pass
Date: Tue Aug 29 00:02:03 2023
InstallationDate: Installed on 2023-08-25 (3 days ago)
InstallationMedia: Ubuntu 22.04.3 LTS "Jammy Jellyfish" - Release amd64 (20230807.2)
SourcePackage: gdm3
UpgradeStatus: No upgrade log present (probably fresh install)
mtime.conffile..etc.gdm3.custom.conf: 2023-08-28T23:54:05.623349