Cyclic sieving conjecture for promotion on fans of Dyck paths

Let $D_n^{(r)}$ be the set of all $r$-fans of Dyck paths of length $2n$, and define

$$g_{n,r}(q)=\prod_{1\leqslant i\leqslant j\leqslant n-1}\frac{[i+j+2r]_q}{[i+j]_q},$$

where $[m]_q=1+q+q^2+\cdots+q^{m-1}$. Let $C_{2n}$ be the cyclic group of order $2n$ acting on $D_n^{(r)}$ by promotion. Cyclic sieving conjecture for fans of Dyck paths. The triple $(D_n^{(r)},C_{2n},g_{n,r}(q))$ exhibits the cyclic sieving phenomenon. The conjecture concerns the interaction between promotion on fans of Dyck paths and the $q$-analogue of their enumeration; the source gives examples but no general proof.