CMS₀ non-definability conjecture for infinite-group gain-graphic matroids

Let $H$ be an infinite group. Let ${\mathit{CMS}_{0}}$ denote counting monadic second-order logic over matroids without quantification over edge sets. Non-definability conjecture. The class of $H$-gain-graphic matroids cannot be characterised with a sentence in ${\mathit{CMS}_{0}}$. The paper notes that existing techniques settle this when $H$ contains elements of arbitrarily high order; the remaining open case is that of infinite groups with finite exponent.