Burns-style congruence conjecture for Rubin–Stark units

Assume $S=S'$, $r=r_{L,S}$, and $r'=r_{L',S}$, and retain the source's definitions of $I_{L'/L}^d$, $Q_{L'/L}^d$, $\varphi_v$, and the induced map $\Phi^{G(L'/L)}$. For every

$$\Phi\in\bigwedge_{\mathcal G_{L'}}^{r'}\operatorname{Hom}_{\mathcal G_{L'}}(\mathcal O_{L',S,T}^{\times},\mathbb Z[\mathcal G_{L'}]),$$

Burns-style congruence conjecture.

$$\Phi(\varepsilon_{L',S,V'})\in I_{L'/L}^d$$

and

$$\Phi(\varepsilon_{L',S,V'})=\operatorname{sgn}(V,V')\Phi^{G(L'/L)}\left(\left(\bigwedge_{v\in V\setminus V'}\varphi_v\right)(\varepsilon_{L,S,V})\right)\quad\text{in }Q_{L'/L}^d.$$

The formulation is described as a slight modification of a theorem of Burns; the source does not establish it in general, although it notes a special unramified case for the related conjecture.