Thomas–Williams' cyclic sieving conjecture for rational Tamari lattices

Let $a$ and $b$ be relatively prime positive integers, let $\operatorname{Tam}(a,b)$ be the rational Tamari lattice, and let $\operatorname{Row}$ be its rowmotion operator. Define the $q$-integer $[k]_q=(1-q^k)/(1-q)$ and the $q$-binomial coefficient ${{a+b\brack b}_q}$ in the usual way. Set

$$\operatorname{Cat}^{(a,b)}(q)=\frac{1}{[a+b]_q}{a+b\brack b}_q.$$

Thomas–Williams' cyclic sieving conjecture. The triple

$$(\operatorname{Tam}(a,b),\operatorname{Row},\operatorname{Cat}^{(a,b)}(q))$$

exhibits the cyclic sieving phenomenon.

This conjecture strengthens the rowmotion-order conjecture by predicting the complete orbit structure through evaluations of the rational Catalan polynomial. It is proved in the paper for $m$-Tamari lattices, but remains open for arbitrary rational Tamari lattices.