Nonnegative order polynomial coefficients for width-two posets

Let $\lambda/\mu\subset [m]\times[n]$ be a skew shape encoding a width-two poset $R_{\lambda/\mu}$, with $m+n$ elements. Width-two positivity conjecture.

$$(m+n)!\cdot\Omega(R_{\lambda/\mu};t)\in\mathbb{Z}_{\geq 0}[t].$$

Thus the normalized order polynomial has nonnegative coefficients. The conjecture is supported in the source by calculations for chains of sizes $m,n\leq5$.