Non-definability conjecture for infinite-group gain-graphic matroids

Let $H$ be an infinite group. A gain-graphic matroid is a frame matroid obtained from a graph with edge gains in $H$, and a class is characterised by a sentence in monadic second-order logic when membership is exactly described by such a sentence. Infinite-group gain-graphic non-definability conjecture. The class of $H$-gain-graphic matroids cannot be characterised by a sentence in monadic second-order logic. Together with the finite-group excluded-minor conjecture, this would give a finite-versus-infinite dichotomy for logical definability of gain-graphic matroid classes; the source records the conjecture as unresolved only for infinite groups of finite exponent.