Selmer-corank conjecture for elliptic curves over the rationals

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $r_{\rm an}(E/\mathbb{Q})$ be the order of vanishing of $L(E/\mathbb{Q},s)$ at $s=1$. For a prime $p$, let $r_p(E/\mathbb{Q})$ denote the $\mathbb{Z}_p$-corank of the Selmer group $\operatorname{Sel}_{p^\infty}(E/\mathbb{Q})$. Selmer-corank conjecture. If

$$r_{\rm an}(E/\mathbb{Q})\geq 2,$$

then

$$r_p(E/\mathbb{Q})\geq 2.$$

This is a weaker Selmer-theoretic version of the preceding Mordell–Weil rank conjecture; the paper explains that the Selmer corank is at least the Mordell–Weil rank, with equality linked to finiteness of the $p$-primary Shafarevich–Tate group.