Generalized Kasteleyn theorem for planar non-bipartite graphs in the disk

Let $\mathcal G$ be a graph in the broader class considered by the cited work, with edge set $\mathcal E$, and let $\sigma:\mathcal E\mapsto\{\pm1\}$ be a signature. For every finite face $\Omega$, write

$$\sigma(\Omega)=\prod_{e\in\partial\Omega}\sigma(e)=(-1)^{n_w(\Omega)+1},$$

where $n_w(\Omega)$ is the number of internal white vertices bounding $\Omega$. Generalized Kasteleyn theorem conjecture. A signature satisfying this face condition realizes the variant of Kasteleyn's theorem for planar non-bipartite graphs in the disk: Pfaffians of minors of the sign matrix count dimer configurations involving every internal vertex exactly once, with prescribed boundary conditions. The usual bipartite correspondence is established earlier in the source, whereas this non-bipartite extension is presented as conjectural and is deferred to future work.