The rational Tamari down-degree homomesy conjecture

Let $a$ and $b$ be relatively prime positive integers. Let $\operatorname{Tam}(a,b)$ be the rational Tamari lattice, with rowmotion operator $\operatorname{Row}$. For $x\in\operatorname{Tam}(a,b)$, define the down-degree statistic by

$$\operatorname{ddeg}(x)=\lvert\{y\in\operatorname{Tam}(a,b):y\lessdot x\}\rvert.$$

Rational Tamari down-degree homomesy conjecture. The statistic $\operatorname{ddeg}$ is homomesic for rowmotion, with average

$$\frac{(a-1)(b-1)}{a+b-1}.$$

The paper proves the analogous statement for $m$-Tamari lattices. The corresponding assertion for general rational Tamari lattices remains open.