Burns's conjecture on higher-rank Stickelberger annihilators

Let $K/k$ be a finite abelian extension of number fields with Galois group $G$, let $S$ be a finite set of places of $k$ satisfying Hypothesis StarkHhigher, and let $S_r=\{v_1,\ldots,v_r\}$. Let $E_S(K)$ and $X_S(K)$ be the corresponding $\mathbb{Z}[G]$-modules, let $\lambda_{K,S}$ be the regulator map, and let $\theta_{K,S}^{*}(0)$, $e_{r,S}'$, $w_K$, and $Cl_S(K)$ have their meanings in the source. For every $\phi\in\operatorname{Hom}_{\mathbb{Z}[G]}(E_S(K),X_S(K))$, set

$$\Theta(\phi)=w_K\theta_{K,S}^{*}(0)e_{r,S}'\operatorname{det}_{\mathbb{C}[G]}(\lambda_{K,S}^{-1}\circ\phi_{\mathbb{C}}).$$

Burns's conjecture. One has $\Theta(\phi)\in\mathbb{Z}[G]$, moreover $\Theta(\phi)\in\operatorname{Ann}_{\mathbb{Z}[G]}(Cl_S(K))$, and if $S'$ satisfies $S_\infty\cup S_r\subseteq S'\subseteq S$, then for every

$$b\in\bigcup_{v\in S\smallsetminus S'}\operatorname{Ann}_{\mathbb{Z}[G]}(\mathbb{Z}[G/G_v])$$

one has $b\Theta(\phi)\in\operatorname{Ann}_{\mathbb{Z}[G]}(Cl_{S'}(K))$. This is a higher-rank refinement of Brumer-type annihilation conjectures, predicting both integrality and annihilation of $S$- and modified class groups. The source specializes Burns's conjecture to the abelian setting and gives numerical evidence, but does not state a resolution.