Demonet's conjecture on rational completeness of the g-vector fan

Let $A$ be a finite-dimensional algebra. Its $\tau$-tilting finiteness is characterized by the $\bg$-vector fan containing the integral Grothendieck lattice

$$\K_0^{\operatorname{proj}}(A)_\bZ \subseteq \fan(A).$$

Demonet's conjecture. $A$ is $\tau$-tilting finite if and only if

$$\K_0^{\operatorname{proj}}(A)_\bZ \subseteq \fan(A).$$

This asks whether rational completeness of the $\bg$-vector fan suffices for $\tau$-tilting finiteness. The supplied source gives no resolution status.