The DO and Schur-representation finiteness conjectures for algebras

Let $\Lambda$ be an algebra. The properties of being DO, Schur-representation-finite, and $\tau$-tilting finite are considered.

DO and Schur-representation finiteness conjectures. The following hold:

1. If $\Lambda$ is DO, then it is $\tau$-tilting finite.
2. $\Lambda$ is Schur-representation-finite if and only if $\Lambda$ is $\tau$-tilting finite.

These implications are established for minimal representation-infinite non-distributive and biserial algebras, and for further specified families, but remain conjectural in general.