Kimoto–Osburn's congruence conjecture for $\tilde{J}_2$

Let $p$ be an odd prime, and let $m,r\geq1$. Let $\tilde{J}_2(n)$ denote the normalized higher Apéry-like number used in the paper, and let $\left(\!\frac{-1}{p}\!\right)$ be the Legendre symbol.

Kimoto–Osburn conjecture. For any odd prime $p$ and any $m,r\geq1$,

$$\tilde{J}_2(mp^r-1)\equiv\left(\!\frac{-1}{p}\!\right)\tilde{J}_2(mp^{r-1}-1)\pmod{p^r}.$$

The source notes that the case $r=1$ follows from elementary binomial congruences and a result of Mortenson, while the general assertion is presented as a conjecture.