Planar simple-matching domination conjecture

Let $G$ and $H$ be graphs with planar union, let $\mathcal{PM}(G,H)$ be the matchings pairing edges from $G$ and $H$, and let $s_M$ be the leading coefficient obtained by summing $(-1)^{\lvert S\rvert}$ over non-crossing subsets $S$ of the relevant vertex set. Planar simple-matching domination conjecture. If $M\in\mathcal{PM}(G,H)$ and $s_M\neq 0$, then there exists a simple matching $M'\in\mathcal{PM}(G,H)$ such that

$$\operatorname{cycles}(M')\geq\operatorname{cycles}(M).$$

This is presented as a concrete combinatorial step toward the planar simple-matching asymptotic conjecture: every uncancelled term should be dominated by a simple matching. The paper does not report a resolution.