Degenerate Frobenius structure on the dual of a G-Euler Frobenius algebra

Let $G$ be a group, let $jin Z(G)$, and let the dual of a $G$-Euler $G$-Frobenius algebra be given. A degenerate $G$-Frobenius algebra of degree $j$ is the structure defined in the surrounding text, with its additional possibly degenerate pairing and degree-$j$ axioms.

Degenerate Frobenius structure conjecture. There is a degenerate $G$-Frobenius algebra of degree $j$ on the dual of a $G$-Euler $G$-Frobenius algebra.

This conjecture proposes a lift of the Frobenius structure expected on the invariant part to a $G$-graded equivariant structure. The supplied text gives no resolution status.