# Theory of representation – Strong conjecture without loop for uniserial modules

Let $$A$$ to be an algebra of Artin. The strong conjecture without a loop says that a simple $$A$$-module with $$Ext_A ^ 1 (S, S) neq 0$$ has an infinite projective dimension.
This assumption has recently been proven for quiver algebras and thus finite dimensional algebras on an algebraically closed field in https://www.sciencedirect.com/science/article/pii/S0001870811002714.

Question: Is an uniserial $$A$$-module $$M$$ with $$Ext_A ^ 1 (M, M) neq 0$$ have an infinite projective dimension?