Application-specific gtk icons do not appear in the global menu.
For example gimp and inkscape have a lot of custom icons that appear if you use the "normal" menu but dont if you use the global menu.

This bug has been bothering me for a long time, today I looked at the code of gmenu-dbusmenu-proxy and I think I realised that it derives the icon to be used from the strings. I wonder if, in collaboration with the developers of appmenu-gtk-module, it might be possible to change this behaviour, perhaps via the file path of the icon used.