The 6-cover conjecture for flow-admissible signed graphs

A signed graph $(G,\Sigma)$ consists of a graph $G$ and a set $\Sigma\subseteq E(G)$ of negative edges. A signed circuit is either a balanced circuit, containing an even number of edges of $\Sigma$, or a barbell, consisting of two unbalanced circuits sharing exactly one vertex or two vertex-disjoint unbalanced circuits joined by a minimal path. The signed graph is flow-admissible if every edge belongs to a signed circuit.

6-cover conjecture. Every flow-admissible signed graph has a family of signed circuits such that every edge belongs to exactly $6$ members of the family.

This extends the circuit-cover problem from ordinary bridgeless graphs to signed graphs. It is known that flow-admissible signed graphs with no $k$-cover exist for every positive integer $k\leq 5$; the conjecture asserts that $6$ is sufficient, while its truth remains open.