Lexicographic shellability conjecture for the uncrossing matching poset

Let $P_n$ be the uncrossing partial order on matchings, and let ${\hat P}_n$ denote its augmented poset. Lexicographic shellability conjecture. The poset ${\hat P}_n$ is lexicographically shellable. The poset is known to be thin, and lexicographic shellability would provide a strong topological property consistent with the expectation that the associated spaces are homeomorphic to balls. The conjecture is presented as widely expected, but no resolution is given here.