  We introduce a new preconditioner for a recently developed pressure-robust
hybridized discontinuous Galerkin (HDG) finite element discretization of the
Stokes equations. A feature of HDG methods is the straightforward elimination
of degrees-of-freedom defined on the interior of an element. In our previous
work (J. Sci. Comput., 77(3):1936--1952, 2018) we introduced a preconditioner
for the case in which only the degrees-of-freedom associated with the element
velocity were eliminated via static condensation. In this work we introduce a
preconditioner for the statically condensed system in which the element
pressure degrees-of-freedom are also eliminated. In doing so the number of
globally coupled degrees-of-freedom are reduced, but at the expense of a more
difficult problem to analyse. We will show, however, that the Schur complement
of the statically condensed system is spectrally equivalent to a simple trace
pressure mass matrix. This result is used to formulate a new, provably optimal
preconditioner. Through numerical examples in two- and three-dimensions we show
that the new preconditioned iterative method converges in fewer iterations, has
superior conservation properties for inexact solves, and is faster in CPU time
when compared to our previous preconditioner.
