Twisted Leopoldt conjecture

Let $F$ be a number field, let $p$ be an odd prime, let $S$ be the set of $p$-adic and infinite primes of $F$, and let $G_S(F)$ be the Galois group of the maximal algebraic extension of $F$ unramified outside $S$. For an integer $i\neq 1$, consider the Galois cohomology group $H^2(G_S(F),\mathbb{Q}_p/\mathbb{Z}_p(i))$. Twisted Leopoldt conjecture. For every number field $F$ and every integer $i\neq 1$,

$$H^2(G_S(F),\mathbb{Q}_p/\mathbb{Z}_p(i))=0.$$

The paper notes that the case $i=0$ is true for abelian extensions of $\mathbb{Q}$ and that all cases $i\geq2$ are known by Soulé; the full assertion as stated is not resolved in general.