Higher-power congruences for Apéry-like sequences

Prove the congruences modulo p^{2r+1} asserted in Conjecture 4.1 for the sequences S_n, a_n, W_n, G_n, G_n^{(3)}, G_n^{(4)}, G_n^{(6)}, and Q_n, relating their indices mp^r-1 and mp^{r-1}-1 to the corresponding Euler-, U-, or s-sequence correction terms for every prime p>3 and all positive integers m and r, with the stated restriction p\ne5 for the G_n^{(4)} congruence.

Background

Section 4 derives several supercongruences for Apéry-like numbers from the identities established earlier in the paper. Conjecture 4.1 proposes extensions of these congruences from the cases treated in the preceding lemmas and theorems to arbitrary positive integers m and r.

The conjecture concerns eight families: the sequence S_n, the sequence a_n, the sequence W_n, and the generalized sequences G_n, G_n^{(3)}, G_n^{(4)}, G_n^{(6)}, and Q_n. Each proposed congruence has modulus p^{2r+1}; the correction terms involve Euler numbers E_{p-3}, the auxiliary sequence U_{p-3}, or the auxiliary sequence s_{p-3}, depending on the family. The sequence s_n is defined immediately after the displayed conjecture, while U_n is defined earlier in Section 4.