Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ a prime. It is conjectured in the book of Coates and Sujatha "Galois Cohomology of Elliptic Curves" (Conjecture 2.5) that the corank of the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$ extension is $1$ when $E$ has potentially supersingular reduction at $p$ and $0$ otherwise. There is a really nice proof of this following Theorem 2.14 which is subsequently given in the case in which the Selmer group of $E$ over $\mathbb{Q}$ is finite (so in particular only when $E$ has rank zero and the $p$ part of the Tate Shafarevich group is assumed to be finite).

What is the current state of affairs on this conjecture?