The Second brick-Brauer-Thrall conjecture for finite-dimensional algebras

Let $A$ be a finite-dimensional algebra, and let $\tau$ denote the Auslander–Reiten translation. Call $A$ brick-infinite when it has infinitely many pairwise non-isomorphic bricks, and write $\tau$-tilting infinite for the equivalent condition. For a dimension vector $\mathbf{d}$, let $\operatorname{rep}(A,\mathbf{d})$ be the corresponding representation variety, with its group action whose orbits are isomorphism classes.

Second brick-Brauer-Thrall conjecture. If $A$ is brick-infinite (equivalently, $\tau$-tilting infinite), there is a dimension vector $\mathbf{d}$ such that $\operatorname{rep}(A,\mathbf{d})$ contains infinitely many orbits of bricks.

This is a conceptual brick analogue of the Second Brauer–Thrall theorem. The statement is presented as an open problem; equivalent formulations in terms of Hom-orthogonal modules and brick-discrete algebras are discussed in the paper.