Existence of an abelian group guaranteeing flow reconfiguration connectivity

Let $G$ be a graph, and let $\mathcal{F}(G,A)$ denote its reconfiguration graph whose vertices are nowhere-zero $A$-flows, with adjacency given by changing flow values only on a cycle of $G$. An abelian group is a group whose operation is commutative, and a graph is 2-edge-connected if it has no bridge. The abelian-group flow reconfiguration conjecture. There is an abelian group $A$ such that, for every 2-edge-connected graph $G$, the reconfiguration graph

$$\mathcal{F}(G,A)$$

is connected. This is a group-flow variant of the integer-flow reconfiguration questions above and remains open.