The tau-tilting finiteness and brick-discreteness conjecture

Let $\Lambda$ be an algebra over an algebraically closed field $k$. It is $\tau$-tilting finite if it has only finitely many isomorphism classes of basic support $\tau$-tilting modules, and it is brick-discrete if, for each dimension, there are only finitely many isomorphism classes of bricks.

Tau-tilting finiteness–brick-discreteness conjecture. The following are equivalent:

1. $\Lambda$ is $\tau$-tilting finite.
2. $\Lambda$ is brick-discrete.

Equivalently, $\Lambda$ is brick-infinite if and only if there exists a family $\{M_{\lambda}\}_{\lambda \in k^*}$ of bricks of the same length.

The conjecture connects $\tau$-tilting finiteness with the geometry and parameter families of bricks. The source presents it as open and as a precursor to the stronger generic-brick conjecture.