The $S$-torsion conjecture for non-commutative Iwasawa modules

Let $p\neq 2$ be a prime, let $F$ be a totally real number field, and let $F^{\infty}/F$ be a totally real Galois extension of infinite degree containing the cyclotomic $\mathbb{Z}_p$-extension, with compact $p$-adic Lie Galois group $G$ and only finitely many ramified primes. Let $\Sigma$ contain all primes ramified in $F^{\infty}$, let $C=C_{F^{\infty}/F}$ be the dual étale cohomology complex, and let $X_{\Sigma}(F^{\infty}/F)$ be the Galois group of the maximal abelian pro-$p$ extension of $F^{\infty}$ unramified outside $\Sigma$. Write $S$ for the canonical Ore set used to define $K_0(\mathbb{Z}_p[[G]],\mathbb{Z}_p[[G]]_S)$. The $S$-torsion conjecture. The class $[C]$ always belongs to $K_0(\mathbb{Z}_p[[G]],\mathbb{Z}_p[[G]]_S)$; equivalently, $X_{\Sigma}(F^{\infty}/F)$ is $S$-torsion. This condition is equivalent, after passage to a suitable finite intermediate field, to the vanishing of the corresponding cyclotomic $\mu$-invariant; the conjecture is known under the stated sufficient $\mu=0$ condition but is not established in general.