A supercongruence relating two cubic binomial sums

Let $p$ be an odd prime and $r$ a positive integer. Define

$$S_{p,r}(x)=\sum_{k=0}^{p^r-1}x^k\sum_{j=0}^k {2j\choose j}^3{2k-2j\choose k-j}^3(3j+1)(3k-3j+1).$$

The supercongruence.

$$S_{p,r}\left(\frac1{16}\right)-S_{p,r}\left(-\frac18\right)\equiv0\pmod{p^{2r+2}}.$$

This strengthens the two preceding congruences, each of which has modulus $p^{r+2}$, by asserting extra cancellation in their difference.