critical situation

You defined some access mode "superviseur" and some profile with same name "superviseur".
To understand, please show access rights to screen Project for profile "Superviseur"

Sorry, was not clear, what I need to see is screen "Accès aux données (dépendant du projet)".
Just one print showing colomn "Superviseur" and line "Projets"

I cannot reproduce with configuration looking like yours.
He is what I did (On Demo).

Create new profile Superviseur


Create new Access Mode Superviseur


Set access to forms to Superviseur profile


Set access to data with access mode Superviseur to Superviseur profile


In workflow default, define access so that Superviseur profile has rights to change status


Create user superviseur with profile Superviseur (in fact it already existed, I just changed profile)


Create new project and allocate Superviseur to this project


Then connect as superviseur,
Go to new project,
And change Status


So sure all works fine.
You can try on demo.projeqtor.org, with user superviseur/superviseur.