The mode insert will stay lit once played. You don't have to actually complete it. The Completed All Scenes insert in the scarf should only light when you gone through all the modes. LEDs can ghost in specific inserts at specific times in the game. It might be related to another insert being lit or just part of the game. Ghosting is where there is some voltage on the circuit that will light an LED because it uses less voltage than an incandescent. You can quickly determine if it's a ghosting issue by trying an incandescent bulb or non ghosting LED.
I've run into ghosting problems like that where it is only one or two inserts. On a couple of occasions, my Shadow ghosts and flickers real bad on a bunch of inserts. It's weird because I will swap one bulb for an incandescent and the problem goes away. I can then put back in the same LED and everything will still be fine.