The even-odd-drop and D-cycle correspondence conjecture

Let $A$ be a finite subset of $\mathbb{Z}_{>0}$. A D-cycle is the cycle notion used in the paper, and an even-odd drop is a drop $(i,\sigma(i))$ with $i>\sigma(i)$, $i$ even, and $\sigma(i)$ odd. Let $\Pi_{\Gamma_{2n}}$ be the partition lattice object used in the paper, let $\mu_{\Pi_{\Gamma_{2n}}}$ be its Möbius function, and let a permutation's cycle support be the partition formed by its cycles.

Even-odd-drop and D-cycle correspondence conjecture. For every finite subset $A\subset\mathbb{Z}_{>0}$, the number of cycles on $A$ with only even-odd drops is equal to the number of D-cycles on $A$. Consequently, for every $n\geq 1$: (i) if $\pi\in\Pi_{\Gamma_{2n}}$, then $|\mu_{\Pi_{\Gamma_{2n}}}(\hat 0,x)|$ equals the number of permutations on $[2n]$ with only even-odd drops and cycle support $\pi$; and (ii) for every $j$, the number of permutations on $[2n]$ with $j$ cycles and only even-odd drops equals the number of D-permutations with $j$ cycles.

The source presents this statement as a conjecture and says that, together with a cited theorem, it would imply both the median Genocchi-number interpretation and the preceding Genocchi cycle-counting conjecture. The supplied text gives no resolution; the precise definitions of D-cycles, D-permutations, and $\Pi_{\Gamma_{2n}}$ should be checked in the paper.