Weak twisted Leopoldt conjecture for multiple Z_p-extensions

Let $F$ be a number field, let $L$ be a $\mathbb{Z}_p^d$-extension of $F$, and let $G_S(L)$ be the Galois group of the maximal extension of $L$ unramified outside the primes above $p$ and the infinite primes. For an integer $i$, weak twisted Leopoldt conjecture. If $i\neq 1$ or $F^c\subset L$, then

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

This is presented as a weak version of the twisted Leopoldt conjecture for arbitrary multiple $\mathbb{Z}_p$-extensions; the source provides no resolution status.