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?