  Let $L/K$ be a finite Galois extension of number fields with Galois group
$G$. Let $p$ be an odd prime and $r>1$ be an integer. Assuming a conjecture of
Schneider, we formulate a conjecture that relates special values of equivariant
Artin $L$-series at $s=r$ to the compact support cohomology of the \'etale
$p$-adic sheaf $\mathbb Z_p(r)$. We show that our conjecture is essentially
equivalent to the $p$-part of the equivariant Tamagawa number conjecture for
the pair $(h^0(\mathrm{Spec}(L))(r), \mathbb Z[G])$. We derive from this
explicit constraints on the Galois module structure of Banaszak's $p$-adic wild
kernels.
