Minuscule P-partition cyclic sieving conjecture

Let $P$ be a minuscule poset, let $r(P)$ denote its rank, and set $h=r(P)+2$. For $\ell\geq 1$, let $\mathrm{PP}^{\ell}(P)$ be the $P$-partitions of height $\ell$, let $\mathrm{row}$ be piecewise-linear rowmotion, and define

$$F(P,\ell)=\prod_{p\in P}\frac{1-q^{r(p)+\ell+1}}{1-q^{r(p)+1}}.$$

Minuscule P-partition cyclic sieving conjecture. For every $\ell\geq 1$, the triple

$$(\mathrm{PP}^{\ell}(P),\langle\mathrm{row}\rangle,F(P,\ell))$$

exhibits the cyclic sieving phenomenon.

This extends the known order-$\ell=1$ result of Rush and Shi and is known for rectangles by work of Rhoades. The conjecture remains open in the supplied text.