On a conjecture concerning the $r$-Euler-Mahonian statistic on permutations

A pair $(\mathrm{st_1}, \mathrm{st_2})$ of permutation statistics is said to be $r$-Euler-Mahonian if $(\mathrm{st_1}, \mathrm{st_2})$ and $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over the set $\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$, where $\mathrm{rdes}$ denotes the $r$-descent number and $\mathrm{rmaj}$ denotes the $r$-major index introduced by Rawlings. The main objective of this paper is to prove that $(\mathrm{exc}_r, \mathrm{den}_r)$ and $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over $\mathfrak{S}_{n}$, thereby confirming a recent conjecture posed by Liu. When $r=1$, the result recovers the equidistribution of $(\mathrm{des}, \mathrm{maj})$ and $(\mathrm{exc}, \mathrm{den})$, which was first conjectured by Denert and proved by Foata and Zeilberger.