Composite-flow coverage conjecture for elementary abelian 2-groups

Let $G$ be a graph, and let $\mathcal{F}(G,\mathbb{Z}_2^k)$ denote the reconfiguration graph of nowhere-zero $\mathbb{Z}_2^k$-flows. A nowhere-zero $\mathbb{Z}_2^k$-flow $f=(f_1,\ldots,f_k)$ is composite if there is a partition $A,B$ of $\{1,\ldots,k\}$ such that, for every edge $e\in E(G)$, there are $i\in A$ and $j\in B$ for which $e$ is in the support of both $f_i$ and $f_j$. The composite-flow coverage conjecture. There is an integer $k\geq 9$ such that, for any 2-edge-connected graph $G$, every connected component of $\mathcal{F}(G,\mathbb{Z}_2^k)$ contains a composite flow. The conjecture proposes that composite flows occur in every reconfiguration component, but its status remains open.