The higher Apéry-like sum congruence for $\tilde{J}_{2k}$

Let $p$ be an odd prime, and let $\tilde{J}_{2k}(n)$ denote the normalized higher Apéry-like number of index $2k$.

Higher Apéry-like sum conjecture. For any odd prime $p$ and any integer $k\geq2$,

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

This is one of the paper's proposed congruence relations among normalized higher Apéry-like numbers; no resolution is given in the source.