If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal. This is one of the many implications of the Birch and Swinnerton-Dyer conjecture.

I want to ask, and excuse me if this is "stupid": Is it enough to show the equality of the ranks for a certain set of primes $p$ to prove the validity of the Tate-Shafarevich conjecture for a given elliptic curve or does one have to prove it for all primes $p$?