Challenging index-multiple congruences

Prove the congruences stated in the challenging conjecture cited from [22] for the values at indices mp^r and mp^{r-1} of S_n, a_n, W_n, G_n, G_n^{(3)}, G_n^{(4)}, G_n^{(6)}, and Q_n, including the displayed correction terms involving the preceding-index values and Euler-, U-, or s-sequence terms modulo p^{2r+1}, with the restriction p\ne5 for the G_n^{(4)} congruence.

Background

Remark 4.1 distinguishes a stronger family of conjectured congruences from the weak version of Conjecture 4.1. It explicitly identifies the statements as a challenging conjecture appearing in reference [22].

The proposed congruences involve the sequences S_n, a_n, W_n, G_n, G_n^{(3)}, G_n^{(4)}, G_n^{(6)}, and Q_n at indices mp^r and mp^{r-1}. Their correction terms use values at m-1 and powers of p, together with Euler numbers, the auxiliary sequence U_n, or the auxiliary sequence s_n; all are asserted modulo p^{2r+1}, subject to the stated exception p\ne5 for G_n^{(4)}.