For abelian variety $A/K$, divisibility problem (i.e. $\forall n≧1$, $Ш(A/K)⊂p^nH^1(G_K,A)$ holds for fixed $p$?) was asked by Cassels in 1962 and even now discussed.

On the other hand, once group structure of $Ш(A/K)$ is known, the question of Cassels holds for enough large $p$. But in my understanding, even the group structure of $Ш(A/K)$ is known, divisibility problem is not finished for small $p$.

My question is, which is more difficult, Cassels divisibility problem or group structure of $Ш(A/K)$ ?

I believe group structure is much more harder question, but does Cassels problem help understanding of group structure of $Ш(A/K)$?