The refined higher Brumer conjecture using Snaith's fractional Galois ideals

Let $K/k$ be an abelian extension of number fields with Galois group $G$, and let $S$ contain the places ramified in $K/k$. For each odd prime $\ell$, let $\mathcal{J}^r(K/k)$ be the $\mathbb{Z}[G]$-submodule of $\mathbb{Q}[G]$ introduced under the higher Stark conjectures. Refined higher Brumer conjecture. For each $r<0$,

$$\operatorname{ann}_{\mathbb{Z}_\ell[G]}(\operatorname{tors}(K_{1-2r}(\mathcal{O}_{K,S}))\otimes_{\mathbb{Z}}\mathbb{Z}_\ell)\,\mathcal{J}^r(K/k)\cap\mathbb{Z}_\ell[G] \subseteq \operatorname{ann}_{\mathbb{Z}_\ell[G]}(K_{-2r}(\mathcal{O}_{K,S})\otimes_{\mathbb{Z}}\mathbb{Z}_\ell).$$

This is presented as a proposed generalization of the higher Brumer conjecture using fractional Galois ideals; the source gives no resolution.