Classification conjecture for bicompatible permutation statistics

A bicompatible permutation statistic is a permutation statistic that is both shuffle-compatible and substring-compatible. Two permutation statistics are considered equivalent when they determine the same classification of permutations in the sense used in the paper.

Classification conjecture. Up to equivalence, the only bicompatible permutation statistics are the descent set $\operatorname{Des}$, the peak set $\operatorname{Pk}$, the valley set $\operatorname{Val}$, and the trivial statistic $\operatorname{triv}$.

This conjecture asks for a complete characterization of the statistics compatible with both shuffles and substrings. It refines the preceding classification claim by including the trivial statistic and considering statistics up to equivalence; the supplied text gives no evidence that it has been resolved.