Gabriel quivers of squared positive-degree quotients

Determine whether, for locally finite standard graded algebras A_ullet and B_ullet over an algebraically closed field k that are isomorphic as ungraded algebras, the finite-dimensional algebras A/A_+^2 and B/B_+^2 necessarily have isomorphic Gabriel quivers.

Background

After dropping semiconnectedness, the paper gives examples where ungraded-isomorphic standard graded algebras need not be graded-isomorphic. The quotients by the squares of the positive-degree ideals retain information about generators and can often be interpreted as quiver-like approximations. The authors ask whether this information is preserved at the level of Gabriel quivers.

References

Let $A_\bullet$ and $B_\bullet$ be locally finite and standard graded, over an algebraically closed field $k$. If $A \simeq B$ as ungraded algebras, do the finite dimensional algebras $A/A_+2$ and $B/B_+2$ have isomorphic Gabriel quivers?

— Isomorphisms of graded semiconnected algebras  (2609.03288 - Dramburg, 3 Sep 2026) in Section 4, Further questions and observations, Question 4.4