Magnitude conjecture for representation-finite bound path algebras

Let $\Lambda=kQ/I$ be a representation-finite bound path algebra of rank $n$, where the rank is the number of isomorphism classes of finitely generated indecomposable projective left $\Lambda$-modules. Write $\chi({\Lambda}\mhyphen\!\operatorname{mod})$ for the magnitude of the module category.

Magnitude conjecture.

1. We have

$$\chi({\Lambda}\mhyphen\!\operatorname{mod})\geq n.$$

2. We have

$$\chi({\Lambda}\mhyphen\!\operatorname{mod})=n$$

if and only if $\Lambda$ is special biserial.

The preceding results establish the equality for several important classes, including representation-finite special biserial algebras, string algebras, Nakayama algebras, and representation-finite local bound path algebras, as well as formulas for Dynkin and certain self-injective cases. The general inequality and the characterization of equality remain open.