Vega's wedge-of-spheres conjecture for matching complexes of caterpillar graphs

Let $G$ be a caterpillar graph, and let $M_k(G)$ denote its $k$-matching complex. Vega's conjecture. The $k$-matching complex $M_k(G)$ is either contractible or homotopy equivalent to a wedge of spheres. This conjecture extends the known result for $1$-matching complexes of forests and Vega's results for $2$-matching complexes of perfect caterpillar graphs; the general case remains open.