The higher Apéry-like square-sum supercongruence for $\tilde{J}_2$

Let $p$ be an odd prime, and let $\tilde{J}_2(n)$ denote the normalized higher Apéry-like number used in the paper. The symbol $\left(\!\frac{-1}{p}\!\right)$ denotes the Legendre symbol.

Square-sum supercongruence. For any odd prime $p$,

$$\sum_{n=0}^{p-1}\tilde{J}_2(n)^2\equiv\left(\!\frac{-1}{p}\!\right)\pmod{p^3}.$$

This conjecture is presented as similar to Rodriguez–Villegas-type congruences for truncated binomial sums; its resolution is not specified in the source.