The folklore conjecture on surjectivity and second cohomology over extensions of the cyclotomic field

Let $p$ be a prime with $p\geqslant 3$, let $F^{cyc}$ be the cyclotomic extension of $F$, and let $F^S$ be the maximal extension of $F$ unramified outside the specified set $S$. For an extension $L$ of $F^{cyc}$ contained in $F^S$, let $\lambda_L$ denote the map in the exact sequence referred to in the source, and let $E_{p^{\infty}}$ denote the $p$-primary torsion subgroup of the elliptic curve $E$. Folklore conjecture. For every such extension $L$, the map $\lambda_L$ is surjective and

$$H^2(F^S/L,E_{p^{\infty}})=0.$$

The vanishing is known for $L=F_{\infty}$, but remains conjectural in general; the restriction $p\geqslant 3$ is necessary because the vanishing can fail for $p=2$.