The sequence $a_n=b^n-c^n$ for $\gcd(b,c)=1$ is a strong divisibility sequence: $m|n\Rightarrow a_m|a_n$, and $a_{\gcd(m,n)}=\gcd(a_m,a_n)$. In particular, $m|n\Leftrightarrow a_m|a_n$. The Lifting the Exponent Lemma states that (with $v_p(N)$ the highest power of an odd prime $p$ dividing $N$)
$v_p(b^n-c^n)=v_p(b-c)+v_p(n)$, if $p|b-c$, $p>2$.
There are similar formulae for $p=2$, and for $v_p(b^n+c^n)$. They are proven using binomial expansions. An elliptic divisibility sequence is a sequence $(a_n)$ that satisfies
$a_{m+n}a_{m-n}a_1^2=a_{m+1}a_{m-1}a_n^2-a_{n+1}a_{n-1}a_m^2, m>n>2$.
If $(a_n)$ satisfies $a_1a_2a_3\neq 0$ (nondegeneracy), $a_1|a_2$, $a_1|a_3$, and $a_2|a_4$, along with $\gcd(a_3,a_4)=1$, then $(a_n)$ is a strong divisibility sequence, similar to $b^n-c^n$.
Question: Is there a Lifting the Exponent lemma for (strong) elliptic divisibility sequences? That is, can we determine $v_p(a_{nm})$ via $v_p(a_{m})$ and $v_p(n)$ (or a different congruence condition on $n$) given that $p|a_{m}$?