Hopkins's rowmotion cyclic sieving meta-conjecture

Let $P$ be a finite graded poset of rank $r$ for which the roots of $\aΩ_P(m)$ are all integers or half-integers, and in the latter case possessing an order two automorphism. Let $m \a0\geq 1$ be an integer and define

$$\Omega_P(m;q):= \prod_{\alpha} \frac{1-q^{\kappa(m-\alpha)}}{1-q^{-\kappa \alpha}},$$

a product over all the roots $\alpha$ of $\Omega_P(m)$ with multiplicity, and with $\kappa:= 1$ if the roots are integers or $\kappa:= 2$ if they are half-integers. Hopkins's rowmotion cyclic sieving meta-conjecture. Then $\Omega_P(m;q)$ is a polynomial in $q$ with nonnegative integer coefficients, and

$$(\mathcal{PP}^m(P),\langle \mathrm{Row} \rangle \simeq \mathbb{Z}/\kappa(r+2)\mathbb{Z},\Omega_P(m;q))$$

exhibits cyclic sieving. This predicts a uniform cyclic sieving phenomenon for rowmotion on $P$-partitions; the paper recalls several proved families and establishes the staircase case, while the general assertion remains open.