Stanton's conjecture on dissections and cyclic sieving polynomials

For a dissection of a polygon, let $f(n,k;q)$ denote the polynomial

$$f(n,k;q):=\frac{1}{[n+k]_q}\left[\begin{array}{c}n+k\k+1\end{array}\right]_q\left[\begin{array}{c}n-3\k\end{array}\right]_q,$$

and for each $\mu\in\mathbb{Z}_{\geq 0}^n$ let $a_\mu(q)$ be the polynomial associated with the class of dissections whose subgon-size multiplicities are given by $\mu$. If $\lambda(\mu)=(\lambda_1>\lambda_2>\cdots>\lambda_k)$ is the partition encoded by $\mu$, define

$$b(\mu):=2k\lambda_1+2(k-1)\lambda_2+\cdots+2\lambda_{k-1}+\lambda_k.$$

Stanton's conjecture. The polynomials $f(n,k;q)$ and $a_\mu(q)$ satisfy

$$q^{2n+k(k-1)}f(n+2,k-1;q)=\sum_\mu q^{b(\mu)}a_\mu(q),$$

where the sum is over all $\mu\in\mathbb{Z}_{\geq 0}^n$ satisfying

$$\sum_{i=1}^n\mu_i=k\qquad\text{and}\qquad\sum_{i=1}^n i\mu_i=n.$$

This conjecture would relate the naive $q$-analogue of the enumeration of polygon dissections to the family of polynomials arising from the cyclic sieving phenomenon; the paper presents it as a relationship conjectured by Dennis Stanton, while the preceding cyclic sieving result establishes the relevant polynomials' fixed-point interpretation.