We show that the compactly supported cohomology of Shimura varieties of Hodge
type of infinite $\Gamma_1(p^\infty)$-level (defined with respect to a Borel
subgroup) vanishes above the middle degree, under the assumption that the group
of the Shimura datum splits at $p$. This generalizes and strengthens the
vanishing result proved in "Shimura varieties at level $\Gamma_1(p^\infty)$ and
Galois representations". As an application of this vanishing theorem, we prove
a result on the codimensions of ordinary completed homology for the same
groups, analogous to conjectures of Calegari--Emerton for completed
(Borel--Moore) homology.