Schneider's conjecture on divisible Galois cohomology

Let $F$ be a global field, let $l$ be a prime with $\operatorname{char} F\ne l$, and, if $l=2$, assume $\mu_4\subset F$. Let $G_S$ be the Galois group of the maximal extension of $F$ unramified outside a finite set $S$ containing the places above $l$, and let $W^n$ denote the relevant $l$-primary Galois module. Define

$$i_n(F)=\dim_{\mathbb F_l}\bigl(\operatorname{Div} H^2(G_S,W^n)[l]\bigr).$$

Schneider's conjecture. $i_n(F)=0$ for all $n\ne 1$. This predicts the vanishing of the divisible part of $H^2(G_S,W^n)$ outside the exceptional twist $n=1$; the supplied text gives the conjectural statement but no resolution status.