Shuffle-compatibility of the up-down run and peak statistics

Let a permutation statistic assign an invariant to each permutation, and call it shuffle-compatible if, for any two disjoint permutations, the multiset of its values on their shuffles depends only on the two input statistics. Let $\operatorname{udr}$ denote the up-down run statistic, $\operatorname{pk}$ the peak statistic, and $\operatorname{des}$ the descent statistic. The up-down run–peak shuffle-compatibility conjecture. The tuples

$$(\operatorname{udr},\operatorname{pk})$$

and

$$(\operatorname{udr},\operatorname{pk},\operatorname{des})$$

are shuffle-compatible. This is one of the paper's empirically supported conjectures about joint permutation statistics; its resolution is not given in the source.