Brumer's conjecture for abelian extensions

Let $L/K$ be an abelian extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in $L/K$. Let $\theta_S$ be the Stickelberger element, let $\mu_L$ be the roots of unity in $L$, and let $\mathrm{cl}_L$ be the class group of $L$. Brumer's conjecture. One has

$$\mathrm{Ann}_{\mathbb{Z}[G]}(\mu_L)\theta_S\subseteq\mathrm{Ann}_{\mathbb{Z}[G]}(\mathrm{cl}_L).$$

This conjecture is an arithmetic integrality and annihilation statement for Stickelberger elements. It is known when $K=\mathbb{Q}$ by Stickelberger's theorem, but remains open in general.