Orbit-structure conjecture for alt $\nu$-Tamari lattices

Let $\nu$ be a lattice path, and let $\mathsf{Tam}_{\delta}(\nu)$ denote the alt $\nu$-Tamari lattice with increment vector $\delta$. Rowmotion acts on this finite lattice, and its orbit structure is the collection of orbit sizes and multiplicities under this action.

Orbit-structure conjecture. The orbit structure of the alt $\nu$-Tamari lattice $\mathsf{Tam}_{\delta}(\nu)$ is independent of the increment vector $\delta$.

The conjecture is supported by the hook and two-row families studied in the paper. The source also states that Adenbaum et al. independently formulated and proved it in forthcoming work, but the supplied status is unknown, so it is recorded as open pending verification of that result.