El Bachraoui's double binomial sum supercongruences

Let $p$ be an odd prime. Define

$$S_-(p)=\sum_{k=0}^{p-1}\left(-\frac{1}{8}\right)^k\sum_{j=0}^k{2j\choose j}{2k-2j\choose k-j}(6j+1)(6k-6j+1)$$

and

$$S_+(p)=\sum_{k=0}^{p-1}\left(\frac{1}{4}\right)^k\sum_{j=0}^k{2j\choose j}{2k-2j\choose k-j}(6j+1)(6k-6j+1).$$

El Bachraoui's conjecture. For every odd prime $p$,

$$S_-(p)\equiv -\frac{p}{2}\pmod{p^2},\qquad S_+(p)\equiv p\pmod{p^2}.$$

These strengthen the corresponding congruences modulo $p$ obtained from earlier $q$-congruences, and are presented as two extensions conjectured by El Bachraoui. The supplied text gives no resolution status.