Rush–Shi's cyclic sieving conjecture for exceptional minuscule posets

Let $P$ be one of the two minuscule posets associated to an exceptional Lie algebra of type $E$, and let $k\in\mathbb{Z}_{\geq 0}$. Write ${\mathsf{PP}}^k(P)$ for the plane partitions of height at most $k$ over $P$, let $\Psi$ denote rowmotion on this set, and let $f_P^k(q)=\sum_{\mathcal{I}\in{\mathsf{PP}}^k(P)}q^{|\mathcal{I}|}$. Rush–Shi's conjecture. The polynomial $f_P^k$ is a cyclic sieving polynomial for the action of $\Psi$ on ${\mathsf{PP}}^k(P)$. This extends the cyclic sieving phenomenon previously known for minuscule plane partitions of height at most $2$ to the exceptional type $E$ cases for every nonnegative height.