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$\DeclareMathOperator\Ext{Ext}$Consider the following statement for a $K$-algebra $A$: $\dim(\Ext^1 (M, N )) \ge \min( \dim(\Ext^1(M, M )), \dim(\Ext^1 (N, N )) ),$ 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.

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(This should probably be a comment but I can't comment yet)

$\DeclareMathOperator\Ext{Ext}$For a class of modules where this inequality holds, one can consider directing modules. Then this inequality holds trivially, since a directing indecomposable module $M$ satisfies $\Ext_A^j(M,M)=0$ for all $j>0$, see Proposition IX.1.4 in Assem, Simson and Skowroński - Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory.

For a class of algebras, one may then consider representation-directed algebras which are algebras where every indecomposable module is directing.

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