>From works of V. A. Abrashkin, we know there exist nontrivial $p$-divisible groups for $p=2$ and some irregular primes.

I don't know where you got this misconception, but it is not true, and Tate's conjecture is very much still expected to hold. That said, the only known general cases are those which proceed via discriminant bounds, which might just include $p = 2$ and $p = 3$.