Eventual p-adic periodicity conjecture for generalized j-Euler numbers

Let $p$ be a prime number, let $m$ be a divisor of $p-1$ or let $m=2$, and let $q$ be the least common multiple of $2$ and $p-1$. Let $j$ be the parameter in the generalized $j$-Euler numbers $\{\mathcal{E}_n^{(d,j)}\}$, defined by

$$\sum_{n=0}^{\infty}\mathcal{E}_{n}^{(d,j)}\frac{t^n}{n!}=\left(1+\sum_{l=1}^{\infty}\frac{t^{dl}}{(dl+j)!}\right)^{-1}.$$

Eventual p-adic periodicity conjecture. The numbers $\mathcal{E}_{mpn}^{(mp,j)}$ are $p$-adic integers, and for every positive integer $r$ there is a positive integer $n_0$ such that

$$\mathcal{E}_{mpn+qp^r}^{(mp,j)}\equiv\mathcal{E}_{mpn}^{(mp,j)}\pmod{p^r}$$

for every $n\geq n_0$. This conjecture predicts eventual $p$-adic periodicity for a composite-parameter family of generalized Euler numbers. The paper reports computational evidence but the supplied text gives no resolution.