Every shuffle-compatible permutation statistic is a descent statistic

Let a permutation statistic be a statistic on permutations, and call it shuffle-compatible if the multiset of its values on shuffles of two permutations depends only on the two input values. A descent statistic is a statistic determined by the descent set, or equivalently by descent data of the permutation as specified in the paper. Descent-statistic conjecture for shuffle-compatible statistics. Every shuffle-compatible permutation statistic is a descent statistic. This conjecture asks whether shuffle-compatibility can occur only for statistics arising from descents. The preceding discussion notes that several non-descent statistics are not shuffle-compatible, but no resolution of the general assertion is supplied.