Consider the following statement for a $K$-algebra $A$: $dim(Ext^1 (M, N )) ≥ min( dim(Ext^1(M, M )), dim(Ext^1 (N, N )) ),$ for all finite dimensional indecomposable $A$-modules $M,N$.

The truth of this inequality has some nice consequences/corollaries.

Question: Is there a nice class of (finite dimensional) algebras $A$ where this inequality holds (maybe for a special class of modules only)? In particular, does this inequality always hold for representation-finite string algebras?

In joint work with Apolonia Gottwald, we proved that the inequality holds for Nakayama algebras, but it feels like it might hold in much larger generality.