Demonet's conjecture on tau-tilting finiteness and E-finiteness

Let $\Lambda$ be a finite-dimensional algebra. It is $\tau$-tilting finite if it has finitely many isomorphism classes of basic $2$-term silting complexes in $K^b(\operatorname{proj}\Lambda)$. It is $E$-finite if every element of $K_0(\operatorname{proj}\Lambda)$ is rigid, meaning that it is the class of a $2$-term presilting complex. Demonet's conjecture. The algebra $\Lambda$ is $\tau$-tilting finite if and only if it is $E$-finite. The implication from $\tau$-tilting finiteness to $E$-finiteness is proved in the paper, so the converse is the remaining open direction.