Congruences involving S_k^(2)(2) and S_k^(2)(3)

Prove the families of mod-p^2 congruences and p-adic integrality assertions involving the specializations S_k^{(2)}(2) and S_k^{(2)}(3) stated in Conjecture 4.6.

Background

This conjecture proposes several related supercongruences for the polynomial family at 2 and 3, with cases determined by representations by x^2+2y^2 and corresponding weighted identities.