The stable second brick-Brauer-Thrall 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 $N\subsetneq M$. Call $A$ brick-infinite if it has infinitely many bricks up to isomorphism. Stable second brick-Brauer-Thrall conjecture. If $A$ is brick-infinite, then there exists some $\theta\in K_0(\operatorname{proj} A)$ such that there are infinitely many $\theta$-stable modules of the same dimension. This is presented as a stronger version of the second brick-Brauer-Thrall conjecture; the source gives no general resolution.