Brumer–Stark conjecture for ray class groups

Let $L/K$ be an abelian CM extension of number fields, with $K$ totally real, and let $G=\operatorname{Gal}(L/K)$. For finite sets of places $S,T$ of $K$, assume $S$ contains all ramified and infinite places, $S\cap T=\varnothing$, and the group $E_L^T$ of units congruent to $1$ modulo the primes above $T$ is torsionfree. Let $\operatorname{cl}_L^T$ be the associated ray class group and let $\theta_S^T$ be the modified Stickelberger element. Brumer–Stark conjecture. For every such pair $S,T$,

$$\theta_S^T\in \operatorname{Ann}_{\mathbb{Z}[G]}(\operatorname{cl}_L^T).$$

This strengthens Brumer's conjecture from ideal class groups to ray class groups. The source presents it as a conjecture of Tate and does not state a general resolution.