Finite-group gain-graphic excluded-minor conjecture

Let $H$ be a finite group, and let $\mathcal{M}$ be any minor-closed class of $H$-gain-graphic matroids. A gain-graphic matroid is a frame matroid obtained from a graph whose edges are assigned elements of $H$. Finite-group gain-graphic excluded-minor conjecture. The class $\mathcal{M}$ has only finitely many excluded minors, and can hence be defined in monadic second-order logic. This is an analogue for gain-graphic matroids of the finite-field finiteness phenomenon for excluded minors and logical definability.