Kimoto–Wakayama's supercongruence for Apéry-like numbers

Let $\tilde{J}_{2}(n)$ be the Apéry-like numbers defined by

$$\tilde{J}_{2}(n):=\sum_{k=0}^{n}(-1)^k\binom{-\frac12}{k}^2\binom{n}{k}.$$

Kimoto–Wakayama's conjecture. For every prime $p\geq 3$,

$$\sum_{k=0}^{p-1}\tilde{J}_{2}(k)^2\equiv(-1)^{\frac{p-1}{2}}\pmod{p^3}.$$

This is a supercongruence for the Apéry-like numbers associated with the non-commutative harmonic oscillator. The conjecture is attributed to Kimoto and Wakayama; its resolution is not indicated in the supplied source.