Refined higher norm formula for Rubin–Stark units

Let $(L,S,V),(L',S',V')\in\Omega$ with $L\subset L'$, $S\subset S'$, and $V\supset V'$. Put $r=|V|$, $r'=|V'|$, and $d=r-r'$. Let $\varphi_v:L^\times\to Q_{L'/L}^1$ be the local-reciprocity map defined in the source for $v\in V$, let $\operatorname{sgn}(V,V')$ be the sign determined by the ordered exterior products, and let $i$ be the canonical injection. Refined higher norm conjecture. The higher norm descent conjecture holds, and

$$i^{-1}\bigl(N_{L'/L}^{(r',d)}(\varepsilon_{L',S',V'})\bigr)=\operatorname{sgn}(V,V')\left(\prod_{v\in S'\setminus S}(1-\operatorname{Fr}_v^{-1})\right)\left(\bigwedge_{v\in V\setminus V'}\varphi_v\right)(\varepsilon_{L,S,V}).$$

This is the main refined compatibility prediction for Rubin–Stark units under extension and change of auxiliary data; the source gives no general resolution.