Nonnegative order polynomial coefficients for cylindric skew shapes

Let $P_{\lambda/\mu/d}$ be the cell poset associated with a cylindric skew shape $\lambda/\mu$ of period parameter $d$, and let $\Omega(P_{\lambda/\mu/d};t)$ be its order polynomial. Cylindric positivity conjecture.

$$|\lambda/\mu|!\cdot\Omega(P_{\lambda/\mu/d};t)\in\mathbb{Z}_{\geq0}[t].$$

The claim is motivated by calculations using the cylindric plane-partition formula; no general proof is given in the source.