Equality of the configuration varieties for finite representation type algebras

Let $\Lambda$ be a finite representation type algebra, and let $\widetilde{\mathcal{U}}_\Lambda$ and $\widetilde{\mathcal{M}}_\Lambda$ be the affine varieties associated with $\Lambda$ described in the paper. Equality conjecture. For any finite representation type algebra $\Lambda$,

$$\widetilde{\mathcal{U}}_\Lambda=\widetilde{\mathcal{M}}_\Lambda.$$

The varieties $\widetilde{\mathcal{M}}_\Lambda$ generalize affine varieties introduced for Dynkin cases, and the paper proves the inclusion $\widetilde{\mathcal{M}}_\Lambda\subseteq\widetilde{\mathcal{U}}_\Lambda$ as well as equality when the Auslander–Reiten quiver of $K_\Lambda$ has no oriented cycles. Examples suggest that equality holds more generally, but the assertion for every finite representation type algebra remains open.