Referee's supercongruence for a truncated double sum with alternating powers of 8

Let $p$ be an odd prime. Define the truncated double sum

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

The first supercongruence conjecture. For every odd prime $p$,

$$S_p\equiv -\frac{p}{2}\pmod{p^2}.$$

This conjecture concerns a supercongruence for a truncated convolution of central binomial coefficients and is suggested by computational evidence. It was proposed by an anonymous referee in connection with the paper's preceding corollaries.