Burns's Fitting ideal conjecture for Selmer modules

Let $H/F$ be a finite abelian CM extension, let $R=\mathbf{Z}[1/2][G]^-$, and define the Selmer module

$$\operatorname{Sel}^T_S(H)=\operatorname{Hom}_{\mathbf{Z}}(H_T^*,\mathbf{Z})/\prod_{w\notin S_H\cup T_H}\mathbf{Z},$$

with contragredient $G$-action. Let $\Theta_{S,T}$ be the smoothed Stickelberger element and let $\#$ denote the involution induced by $g\mapsto g^{-1}$. Burns's conjecture. One should have

$$\operatorname{Fitt}_R(\operatorname{Sel}^T_S(H)^-)=(\Theta_{S,T}^\#).$$

This seeks an arithmetic Selmer module whose Fitting ideal is generated by the Stickelberger element. The source gives no resolution status.