Have you tried pinching together the display and core module in the contact area (that should be in the lower centre-left when you are looking at the screen) that connects them? When you remove the battery and then connect the FP2 to a charger, do you get to hear recurring vibrations (indicating a boot loop)?
P.S.: If there is another FP2 user or a Fairphone Angel nearby, you might also try to temporarily swap core and display modules to test yours.