separate firefox profiles open in the same dock icon

[Linda]

This worked in Ubuntu 20.04, and broke in Ubuntu 22.04.1 LTS.

I'm not sure if the module this bug is in is gnome. This seems to describe a similar issue:

https://bugs.chromium.org/p/chromium/issues/detail?id=118613

except my problem occurs with multiple profiles of firefox, and is not fixed by adding --user-data-dir. The chromium bug report suggests the --class argument is ignored, which may be the problem.

I have a separate icon and separate desktop file for each firefox profile. When I click on one of these icons in the dock, it

(buggy behavior) creates a new icon on the dock with the generic firefox icon, instead of
(expected behavior) associating the newly opened window with my unique icon for each profile.

The chromium bug link above suggests it's fixed by adding --user-data-dir, but I tried that, e.g., --user-data-dir=/tmp/firefox1, but that didn't fix it for me.

My desktop files contain:

firefox1.desktop:
  Exec=firefox -P default %u --class firefox1
  StartupWMClass=firefox1
  wmclass=firefox1
  Icon=firefox-green48.png

firefox2.desktop:
  Exec=firefox -no-remote -P second %u --class firefox2
  StartupWMClass=firefox2
  wmclass=firefox2
  Icon=firefox-purple48.png

firefox3.desktop:
  Exec=firefox -no-remote -P third %u --class firefox3
  StartupWMClass=firefox3
  wmclass=firefox3
  Icon=firefox-yellow48.png