Some description what we're seeing might help, this is a beauty panel that's molded vinyl (for a faux leather look?) with a foamy backing to it, that originally used adhesive to attach to the glovebox door, or was molded onto it?
If it were mine, I'd try acrylic caulking, because I don't know if acid or ammonia cure silicone would react with the vinyl/foam/whatever. I'd use clamps and cardboard to spread the clamp tension till it cured. I'd use a generous amount and since it's acrylic, water washup any excess that squishes out, before it dries... might also be good to pick a color, the same shade or barely lighter color than the piece. Almond maybe? IDK.
Considering it may be open cell foam but a sealed top surface, could take a long time for acrylic caulking to dry. Pointing a fan at it might help.