  We propose a $p$-multilevel preconditioner for Hybrid High-Order
discretizations (HHO) of the Stokes equation, numerically assess its
performance on two variants of the method, and compare with a classical
Discontinuous Galerkin scheme. We specifically investigate how the combination
of $p$-coarsening and static condensation influences the performance of the
$V$-cycle iteration for HHO. Two different static condensation procedures are
considered, resulting in global linear systems with a different number of
unknowns and non-zero elements. An efficient implementation is proposed where
coarse level operators are inherited using $L^2$-orthogonal projections defined
over mesh faces and the restriction of the fine grid operators is performed
recursively and matrix-free. The various resolution strategies are thoroughly
validated on two- and three-dimensional problems.
