Asymptotic homomesy for repeated lattice-path Tamari lattices

Let $\nu_0$ be a lattice path with $a$ north steps and $b$ east steps. For each integer $t\geq 1$, let $\nu_0^t$ be the path obtained by concatenating $t$ copies of $\nu_0$. Let $\operatorname{Tam}(\nu_0^t)$ be the associated $\nu_0^t$-Tamari lattice, let $\operatorname{Row}$ be its rowmotion operator, and let $\operatorname{Orb}_{\operatorname{Row}}(\operatorname{Tam}(\nu_0^t))$ be the set of rowmotion orbits. Define $\operatorname{ddeg}$ as the down-degree statistic.

Asymptotic homomesy conjecture.

$$\lim_{t\to\infty}\max_{\mathcal O\in\operatorname{Orb}_{\operatorname{Row}}(\operatorname{Tam}(\nu_0^t))}\left\lvert\frac{1}{t}\cdot\frac{1}{\lvert\mathcal O\rvert}\sum_{x\in\mathcal O}\operatorname{ddeg}(x)-\frac{ab}{a+b}\right\rvert=0.$$

The conjecture predicts that normalized orbit averages of down-degree become asymptotically independent of the orbit. It is motivated by computational evidence that general $\nu$-Tamari down-degree averages are close, but not equal, across rowmotion orbits.