Coates–Sinnott conjecture for higher K-groups

Let $L/K$ be a finite abelian extension of number fields and let $G=\operatorname{Gal}(L/K)$. Let $r$ be a negative integer, and let $S$ be a finite set of places of $K$ containing all infinite places and all places ramified in $L$. Write $K_n(\mathcal{O}_{L,S})$ for the Quillen $K$-group and let $\theta_S(r)$ be the Stickelberger element. Coates–Sinnott conjecture.

$$\operatorname{Ann}_{\mathbb{Z}[G]}(K_{1-2r}(\mathcal{O}_L)_{\mathrm{tors}})\theta_S(r) \subseteq \operatorname{Ann}_{\mathbb{Z}[G]}(K_{-2r}(\mathcal{O}_{L,S})).$$

This is the higher $K$-theoretic analogue of Brumer's conjecture. The paper derives it away from its $2$-primary part under the stated hypotheses.