  The shuffle product has a connection with several useful permutation
statistics such as descent and peak, and corresponds to the multiplication
operation in the corresponding descent and peak algebras. In their recent work,
Gessel and Zhuang formalized the notion of shuffle-compatibility and studied
various permutation statistics from this viewpoint. They further conjectured
that any shuffle compatible permutation statistic is a descent statistic. In
this note we construct a counter-example to this conjecture.
