It finally works! I used to play with the vanilla GD main menu instead of FG. So I just downloaded the Main Menu Swhitch tool here : [Request] Main Menu Selection - #16 by Danu88 and it solved the problem!
If I can suggest a modification: the almost disappearing window when the mouse cursor is not hovering it is a little annoying to me.
Thanks again for GD MultiStash, a very useful tool!