Esperet et al.'s flow reconfiguration equivalence conjecture

Let $G$ be a graph and $k$ a positive integer. A $k$-flow is a nowhere-zero integer flow with $0<|f(e)|<k$ on every edge, and an $A$-flow is a nowhere-zero flow with values in an abelian group $A$. Let $\mathcal{F}(G,k)$ and $\mathcal{F}(G,\mathbb{Z}_k)$ be the reconfiguration graphs whose vertices are the corresponding flows and whose edges represent flow-adjacency.

Esperet et al.'s flow reconfiguration conjecture. For each graph $G$ and each positive integer $k$,

$$\mathcal{F}(G,k)\text{ is connected if and only if }\mathcal{F}(G,\mathbb{Z}_k)\text{ is connected}.$$

One direction is easy, since reducing flow values modulo $k$ maps walks to walks. The reverse direction remains open; the source also gives examples showing that reconfiguration can distinguish nonisomorphic groups of the same order.