Equality of maximal in-degree and out-degree subposets in ν-Tamari lattices

Let $\nu$ be an arbitrary path. Define $\Dinnu$ and $\Doutnu$ as the subposets of the $\nu$-Tamari lattice induced by elements with maximal in-degree and maximal out-degree, respectively.

Maximal-degree subposet conjecture.

$$\order{\Dinnu}=\order{\Doutnu}.$$

Computations suggest that the numbers of elements with maximal in-degree and maximal out-degree agree. The conjecture proposes that this equality holds for every path $\nu$.