Transpose-and-rowmotion cyclic sieving conjecture for even-height plane partitions

Let $\mathrm{PP}^{2M}(2n\times2n)$ denote plane partitions in a $2n\times2n$ box of height at most $2M$, let $\mathrm{Tr}$ denote transposition, and let $\mathrm{Row}$ denote rowmotion. Set $\zeta=e^{\pi i/(2n)}$, a primitive $(4n)$th root of unity, and

$$F(q)=\prod_{1\leq i,j\leq n}\frac{1-q^{2(i+j+M-1)}}{1-q^{2(i+j-1)}}.$$

Transpose-and-rowmotion cyclic sieving conjecture. For every $k\in\mathbb Z$,

$$\#\{\pi\in\mathrm{PP}^{2M}(2n\times2n):\mathrm{Tr}(\pi)=\pi,\ \mathrm{Row}^{2n}(\pi)=\pi,\ \mathrm{Row}^{k}(\pi)=\pi\}=F(\zeta^k).$$

This is the corner-triangular reformulation in the source; the supplied text gives no resolution.