Narayana refinement conjecture for the sets ${\cal A}_n$

Suppose that $n\ge 2$, and let $D_{nk}$ be the number of elements of ${\cal A}_n$ for which equality holds for $k$ of the relevant inequalities. Let $N(n,k)$ denote the Narayana number

$$N(n,k)=\frac{1}{n}\binom{n}{k}\binom{n}{k-1}.$$

Narayana refinement conjecture. The number $D_{nk}$ is divisible by

$$\prod_{i=0}^{n-3}\frac{1}{i+1}\binom{2i}{i},$$

and the quotient is the Narayana number $N(n-2,k)$.

This is presented as a stronger combinatorial refinement of the volume conjecture, classifying the elements of ${\cal A}_n$ by the number of equalities they satisfy. The paper states that it has not been proved.