Burns's strengthened Rubin–Stark congruence

Let $I$ be the augmentation ideal of $\mathbb{Z}[\Gamma]$, let $\epsilon$ be the element supplied by the Rubin–Stark conjecture, and let $\Phi\in\iota(\Lambda^n U^*)$. Let $Reg_{\Gamma}^{\Phi}$ denote the corresponding equivariant regulator, and let $h_{k,S,T}$ be the modified class number. Burns's strengthened Rubin–Stark conjecture. Assuming the Rubin–Stark conjecture, for every $\Phi\in\iota(\Lambda^n U^*)$ one has $\Phi(\epsilon)\in\mathbb{Z}[\Gamma]$ and

$$\Phi(\epsilon)\equiv h_{k,S,T}Reg_{\Gamma}^{\Phi}\pmod{I^{r_k-n+1}}.$$

This strengthens the integral Rubin–Stark assertion by prescribing the leading augmentation-ideal term through an equivariant regulator. The source does not state whether this strengthened conjecture is resolved.