The p-adic convergence conjecture for elliptic divisibility sequences

Let ${\mathcal W}=(W_n)_{n\ge0}$ be an elliptic divisibility sequence and let $p$ be a prime. For an exponent $N=N_p\ge1$, consider the subsequences indexed by $mp^{kN}$ for each $m\ge1$. The p-adic convergence conjecture. There is an exponent $N=N_p\ge1$ such that for every $m\ge1$, the limit

$$\lim_{k\to\infty} W_{mp^{kN}}$$

converges in $\mathbb{Z}_p$ to a number algebraic over $\mathbb{Q}$. The paper proves the assertion for almost all primes in the non-CM case, and for all but finitely many primes splitting in the CM field in the CM case; the conjecture proposes it for every elliptic divisibility sequence and every prime.