Pull up on the suction cup with firm, constant pressure to create a slight gap between the screen and the frame.

Insert an opening pick into the gap.

The watertight adhesive holding the screen in place is very strong; creating this initial gap takes a significant amount of force. If you're having a hard time opening a gap, apply more heat, and gently rock the screen up and down to weaken the adhesive until you create enough of a gap to insert your tool.