A (failed?) attempt at the Deligne-Simpson problem

The Deligne–Simpson problem asks for criteria ensuring the existence of an irreducible local system on a punctured projective line with fixed conjugacy classes of local monodromies. After recalling the solutions by Crawley-Boevey and Shu, I discuss a different approach to the necessary conditions in Crawley-Boevey's criterion. The proposed proof contains a gap; the lecture explains where the difficulty occurs and puts the attempted argument in perspective.