Congruences involving S_k^(2)(4)

Prove the mod-p^2 congruences and p-adic integrality relations involving S_k^{(2)}(4) and the associated weighted sums specified in Conjecture 4.4.

Background

The conjecture concerns another specialization of S_k^{(2)} and includes cases governed by representations by x^2+5y^2, as well as higher-power p-adic extensions.