The Genocchi cycle-counting conjecture

A drop of a permutation or cycle on a finite set of positive integers is a pair $(i,\sigma(i))$ with $i>\sigma(i)$; it is an even-odd drop when $i$ is even and $\sigma(i)$ is odd. Let $g_n$ denote the Genocchi numbers.

Genocchi cycle-counting conjecture. For all $n\geq 1$, $g_n$ is equal to the number of cycles on $[2n]$ with only even-odd drops.

This gives a combinatorial formula for the Genocchi numbers analogous to the stated formula for the median Genocchi numbers. The conjecture was verified by computer for $n\leq 6$, but no general proof or resolution is supplied here.