Facial-cycle conjecture for non-decomposable critically frustrated signed plane graphs

Let $(G,\sigma)$ be a non-decomposable critically $k$-frustrated signed plane graph, and call the bounded and unbounded regions determined by its embedding facial cycles when their boundaries are cycles of $G$; a cycle is negative when the product of the signs on its edges is negative.

Facial-cycle conjecture. Every non-decomposable critically $k$-frustrated signed plane graph has exactly $2k$ facial cycles, each of which is a negative cycle.

This is a plane-graph restriction of the finiteness conjecture for $\mathcal{L}^*(k)$ and concerns the structure of critically frustrated signed plane graphs; the source gives no resolution.