Isomorphism under alternative Fermi orientations

Prove or disprove that the double quiver algebras associated with different choices of Fermi-multiplet orientations are isomorphic whenever those orientation choices yield the same partition function.

Background

For theories with two supercharges, Fermi multiplets require an orientation convention when defining the one-loop determinant and the double quiver algebra. Different conventions produce the same partition function, but the corresponding bond factors can differ by signs or branch choices. The paper explicitly conjectures that the resulting algebras are nevertheless isomorphic.

References

Since \widetilde{\phi}{a\Leftarrow b}(z) in different conventions would appear in the relations of the algebra for the same quiver and since different conventions would still give the same partition function, it is natural to conjecture that the algebras would be isomorphic.

— Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues  (2501.03365 - Bao et al., 6 Jan 2025) in Section 7.1, paragraph following equation (7.16)