Gessel and Zhuang's shuffle-compatibility conjecture for $(\mathrm{udr},\mathrm{pk},\mathrm{des})$

For a permutation $\tau$, let $\rm udr$, $\rm pk$, and $\rm des$ denote the permutation statistics of the length of the longest up-down run, the number of peaks, and the number of descents, respectively. A statistic is shuffle-compatible if, for any two permutations with disjoint ground sets, the multiset of its values over their shuffles depends only on the statistic values of the two permutations. Gessel and Zhuang's conjecture. The triple $(\rm udr,\rm pk,\rm des)$ is shuffle-compatible. This conjecture concerns whether the joint distribution of these three permutation statistics is preserved under shuffling in the required sense. The paper states that it proves this conjecture, so the conjecture is resolved.