Levit's matching-cycle conjecture for almost bipartite non-König–Egerváry graphs

Let $G$ be an almost bipartite non-König–Egerváry graph, and let $C$ be its unique odd cycle. A maximum matching of $G$ is a matching with the largest possible number of edges. Levit's matching-cycle conjecture. Every maximum matching in $G$ contains

$$\left\lfloor \frac{\left|V(C)\right|}{2}\right\rfloor$$

edges belonging to $C$.

This strengthens the known result that every maximum matching contains at least one edge of the unique odd cycle; the analogous assertion is not true for almost bipartite König–Egerváry graphs.