Homometry conjecture for down-degree on alt $\nu$-Tamari lattices

Let $\nu$ be a lattice path, let $\delta$ be an increment vector, and let $\mathsf{Tam}_{\delta}(\nu)$ be the corresponding alt $\nu$-Tamari lattice. Under rowmotion, the down-degree statistic assigns to each element the number of elements covered immediately below it, and homometry means that its total over an orbit depends only on the orbit size.

Alt $\nu$-Tamari homometry conjecture. The down-degree statistic is homometric for rowmotion on alt $\nu$-Tamari lattices $\mathsf{Tam}_{\delta}(\nu)$ and is independent of the increment vector $\delta$.

The hook and two-row families studied in the paper support this extension of the corresponding conjecture for ordinary $\nu$-Tamari lattices. Whether the assertion holds for all lattice paths and increment vectors is not resolved in the supplied text.