Luschny's permanent conjecture for median Genocchi numbers

Let $n$ be a positive integer and define the $2n\times 2n$ matrix

$$M_{2n}:=\left[\left\lfloor\frac{2j-k-1}{2n}\right\rfloor\right]_{1\le j,k\le 2n}.$$

The Luschny permanent conjecture.

$$\operatorname{per}(M_{2n})=(-1)^nH_{2n-1}.$$

Here $H_1,H_3,H_5,\ldots$ are the Genocchi numbers of the second kind, or median Genocchi numbers. The conjecture was proposed by P. Luschny and is one of the five permanent conjectures proved in the paper, so it is solved.