[Refugee]

I would not be tempted with the iron.
It looks like soft vinyl and is not as durable as regular speaker cloth. The iron may smooth out the texture before it sticks back down.
I would take the back off or even take the deck out to gain access to a bit at the edge that will not matter if it gets damaged and try various methods to soften the existing glue or add some new glue but wait over night to see if it soaks through to the finish before committing to the repair.