Equidistribution of the descent and cycle bistatistics with the excedance bistatistic

Let $S_n$ be the set of permutations of $\\{1,2,\ldots,n\\}$, and let $\operatorname{des}_2$, $\operatorname{cyc}$, $\operatorname{pex}$, $\operatorname{des}$, and $\operatorname{exc}$ denote the statistics defined in the paper. Two bistatistics are equidistributed on $S_n$ when they have the same joint distribution. Equidistribution conjecture. The two bistatistics

$$(\operatorname{des}_2,\operatorname{cyc})$$

and

$$(\operatorname{pex},\operatorname{cyc})$$

are equidistributed on $S_n$. This conjecture proposes a joint refinement of the preceding equidistribution results for these permutation statistics; its resolution is not given in the supplied text.