Hopkins–Rubey cyclic sieving conjecture for linear extensions of V(n)

Let $V(n)$ be the poset in the source, let $\mathcal{L}(V(n))$ denote its set of linear extensions, and let $\operatorname{Pro}$ denote promotion. Hopkins–Rubey's conjecture. For every $n\geq 1$, the rational expression

$$f(q)=\frac{\prod_{i=1}^{3n}(1-q^{2i})}{\prod_{i=2}^{2n+1}(1-q^i)\prod_{i=2}^{n+1}(1-q^{2i})}$$

is in $\mathbb{N}[q]$, and $(\mathcal{L}(V(n)),\langle\operatorname{Pro}\rangle,f(q))$ exhibits cyclic sieving.

The conjecture concerns both polynomiality with nonnegative coefficients and the orbit structure of promotion; the source does not report a resolution.