Determine whether Hom-Ext relations are always monomial

Determine whether, for the Hom-Ext quiver of an exceptional collection over an arbitrary finite acyclic quiver, all defining relations are generated by monomial relations, or whether non-monomial relations can occur.

Background

For quivers of type $\widetilde{\mathbb A}$, the paper proves that the Hom-Ext quiver algebra is gentle, so its relations are generated by monomials of length two. The authors ask whether this phenomenon extends beyond the affine type $\widetilde{\mathbb A}$ setting.

The unresolved issue concerns arbitrary finite acyclic quivers: the question is whether monomial relations suffice to describe the Hom-Ext quiver algebra, or whether genuinely non-monomial relations must sometimes be included.