The stable brick conjecture

Let $A$ be an algebra, and let $\theta\in K_0(\operatorname{proj} A)$. A module $M\in\operatorname{mod} A$ is $\theta$-stable if $\theta(\underline{\dim}M)=0$ and $\theta(\underline{\dim}N)<0$ for every nonzero proper submodule $N\subsetneq M$. Stable brick conjecture. If $A$ admits an infinite family of bricks of the same dimension, then for some $\theta\in K_0(\operatorname{proj} A)$, there are infinitely many $\theta$-stable modules of the same dimension. The conjecture is known for all tame algebras and for algebras admitting a preprojective component in their Auslander-Reiten quiver, but remains open in general.