The Hall algebra group homomorphism conjecture for tau-tilting finite algebras

Let $\Lambda$ be an arbitrary $\tau$-tilting finite algebra. Let $G(\Lambda)$ be the group associated with the relations from 2-simple minded collections, and let $\mathcal{H}_{0\textnormal{-PS}}^*(\Lambda)$ be the corresponding Hall algebra. The Hall algebra homomorphism conjecture. There is a group homomorphism

$$\phi_\Lambda:G(\Lambda)\longrightarrow\mathcal{H}_{0\textnormal{-PS}}^*(\Lambda)$$

given on generators by

$$\phi_\Lambda(X_S)=(1-[S])^{-1}.$$

The conjecture extends the theorem proved under the assumption that every pair of Hom-orthogonal bricks is either a pair of $K$-stones or Ext-orthogonal. The paper verifies the result for polygons corresponding to the Dynkin diagrams $B_2\cong C_2$ and $G_2$, while the general case remains open because a classification of all 2-vertex $\tau$-tilting finite algebras is not currently available.