Weak Leopoldt conjecture for a $p$-adic Galois representation

Let $\mathcal{K}$ be a finite extension of $\mathbb{Q}_p$, let $\mathcal{O}$ be its valuation ring, and let $\mathbf{A}(\rho)=\mathbf{V}_\rho/\mathbf{T}_\rho$ be the $p$-divisible module attached to a continuous integral Galois representation $\rho$. Let $K_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of a number field $K$, and let $K_S$ be the maximal extension unramified outside $S$. Weak Leopoldt conjecture. One has

$$H^2(K_S/K_\infty,\mathbf{A}(\rho))=0.$$

This is a standard cohomological vanishing conjecture related to the cotorsionness and $mu$-invariants of fine Selmer groups; the source does not state a general resolution.