Demonet's conjecture on the tau-tilting fan

Let $A$ be an algebra of rank $n$, and let $\mathcal{F}_A$ denote its $\tau$-tilting fan in $\mathbb{R}^n$. Call $A$ brick-infinite if it has infinitely many bricks up to isomorphism. Demonet's conjecture. If $A$ is brick-infinite, there is a ray in $\mathbb{Q}^n$ that does not belong to the $\tau$-tilting fan of $A$. The statement is equivalent to a question posed by Demonet and is open in full generality; it is related to the second brick-Brauer-Thrall conjecture.