Higher norm descent conjecture 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 $d=r-r'$, and let $i$ be the canonical injection from the $d$-th augmentation quotient construction into the target of the higher norm $N_{L'/L}^{(r',d)}$. Higher norm descent conjecture.

$$N_{L'/L}^{(r',d)}(\varepsilon_{L',S',V'})\in\operatorname{Im}i.$$

The claim is true when $d=0$ by the remarks cited in the source and is related to Kolyvagin derivative constructions and Kolyvagin systems; the general case remains open in the paper.