Undecidability conjecture for rank-3 gain-graphic matroids

Let $H$ be an infinite group, and consider the class of rank-$3$ $H$-gain-graphic matroids. Undecidability conjecture. This class has an undecidable ${\mathit{MS}_{0}}$ theory. The source presents this as a belief rather than a proved result; it follows the preceding discussion of the unresolved finite-exponent case for infinite groups.