Maximum-degree conjecture for critically frustrated signed graphs

Let $(G,\sigma)$ be a critically $k$-frustrated signed graph, and let $\Delta(G)$ denote the maximum degree of $G$. A cycle is negative when the product of the signs on its edges is negative.

Maximum-degree conjecture. Every critically $k$-frustrated signed graph $(G,\sigma)$ satisfies

$$\Delta(G)\leq 2k.$$

Moreover, equality $\Delta(G)=2k$ can occur only when $G$ consists of $k$ negative cycles that are pairwise edge-disjoint but all contain the same vertex.

This conjecture proposes both a universal maximum-degree bound and a characterization of the equality case; the source gives no resolution.