Kervaire–Murthy adelic class-group conjecture

Let $F_n$ be the cyclotomic field in question, let ${\mathbb A}(F_n)$ be its ring of adeles, and let $K_{p^{n+1}-1}$ be the subgroup

$$K_{p^{n+1}-1}=GL(1,\mathbb Q)\times U_{n,p^{n+1}-1}\times\prod GL(1,\mathbb Q_w)$$

of $GL(1,{\mathbb A}(F_n))$, where the product ranges over the valuations $w$ distinct from $\mu_n=(1-\zeta_n)$. Kervaire–Murthy conjecture. The conjecture can be formulated as

$$(S_n^-)^*\cong\left(GL(1,F_n)\setminus GL(1,{\mathbb A}(F_n))/K_{p^{n+1}-1}\right)^+_{(p)}.$$

This is an adelic reformulation of the Kervaire–Murthy conjecture, relating the character group of the minus part of the class group to a plus $p$-primary adelic double quotient; the supplied passage gives no separate resolution of this formulation.