Finite-group gain-graphic well-quasi-order conjecture

Let $H$ be a finite group. A gain-graphic matroid is a frame matroid obtained from a graph with edge gains in $H$, and a class is well-quasi-ordered when it has neither an infinite strict descending sequence nor an infinite antichain under the relevant minor relation. Finite-group gain-graphic well-quasi-order conjecture. The class of $H$-gain graphic matroids is well-quasi-ordered. If true, every minor-closed class of $H$-gain-graphic matroids would have at most finitely many excluded minors that are themselves $H$-gain-graphic. The source says this is widely believed and expected to be approachable, at least for abelian $H$, while the corresponding assertion fails for infinite groups.