The ASD-type congruence conjecture for $\tilde{J}_2$

Let $p$ be an odd prime and let $m,r\in\mathbb{Z}_{>0}$ with $m$ odd. Let $\tilde{J}_2(n)$ denote the normalized higher Apéry-like number used in the paper. Define the coefficients $\lambda_n$ by

$$\sum_{n=1}^{\infty}\lambda_nq^n=q\prod_{n=1}^{\infty}(1-q^{4n})^6=\eta(4\tau)^6.$$

ASD-type congruence conjecture. For any odd prime $p$ and $m,r\in\mathbb{Z}_{>0}$ with $m$ odd,

$$\tilde{J}_2\left(\frac{mp^r-1}{2}\right)-\lambda_p\tilde{J}_2\left(\frac{mp^{r-1}-1}{2}\right)+(-1)^{p(p-1)/2}p^2\tilde{J}_2\left(\frac{mp^{r-2}-1}{2}\right)\equiv0\pmod{p^r}.$$

Further, the congruence holds modulo $p^{2r}$ if $p\geq5$. The source says that the Apéry-number congruence is a special case and that the analogous Apéry and normalized higher Apéry-like numbers satisfy the same relation, but it does not state a resolution.