The E-finiteness conjecture for finite-dimensional algebras

Let $A$ be a finite-dimensional algebra. Call $A$ $E$-finite if, for every $\tau$-regular irreducible component $\mathcal{Z}\in\operatorname{Irr}(A)$, one has $c(\mathcal{Z})=0$. Call $A$ brick-finite if it has only finitely many isomorphism classes of bricks.

E-finiteness conjecture. If $A$ is $E$-finite, then $A$ is brick-finite.

The conjecture reformulates a question of Demonet about the existence of rational rays outside the $\tau$-tilting fan for brick-infinite algebras. It is stated as open and is described as widely believed.