Infinite q-log-convexity conjecture for matching crossing-number polynomials

For each $n\geq 1$, let $M_{2n}(x)=\sum_k M_{2n,k}x^k$, where $M_{2n,k}$ is the number of matchings on $[2n]$ with crossing number $k$. A polynomial sequence is infinitely q-log-convex when every iterate of the relevant $q$-log-convexity operator is $q$-log-convex. Infinite $q$-log-convexity conjecture. The polynomial sequence $\{M_{2n}(x)\}$ is infinitely $q$-log-convex. Furthermore, the polynomials $M_{2n}(x)$ are log-concave of any order with respect to infinitely $q$-log-convexity. This is the paper's strongest stated regularity proposal for matching crossing-number polynomials; the supplied text gives no resolution status.