To prove BSD conjecture, one has to know about 'finiteness of Shafarevich Tate group'. But, existence of an example of an elliptic curve with rank 2 (whose Sha group $III(E/\mathbb{Q})$ is finite) is not yet known.

Is there any example of rank 2 elliptic curve such that $p$-primary component of $III$ is trivial for $p$ outside a finite set of primes?. In particular, $III(E/\mathbb{Q})[p]$ is trivial for $p$ $\neq$ 2, 3, 5, 7.