The wedge-of-spheres conjecture for matching complexes of planar graphs

Let $G$ be a planar graph, and let $\mathcal{M}(G)$ denote its matching complex, whose vertices are the edges of $G$ and whose simplices are sets of pairwise disjoint edges. Planar matching-complex conjecture. The matching complex $\mathcal{M}(G)$ is contractible or homotopy equivalent to a wedge of spheres. The result would extend the corresponding classification known for several classes of outerplanar and planar graphs; the paper gives no example of a planar graph whose matching complex has neither of these homotopy types, so the conjecture remains open.