Iwasawa-theoretic Mazur–Rubin–Sano congruence conjecture

Let $L_n/L/k$, $V\subseteq V'$, $r$, $r'$, $e$, $\Gamma_n$, $\nu_n$, and $\mathcal{N}_n$ be as in the finite-level Mazur–Rubin–Sano setup, and let $\operatorname{Rec}_\infty$ be the inverse-limit reciprocity map. Assume the $p$-component Rubin–Stark conjecture for $L_n/k$ and $V$ for every $n$.

Iwasawa-theoretic Mazur–Rubin–Sano conjecture. There exists a unique

$$\kappa=(\kappa_n)_n\in\bigcap^rU_{L,S,T}\otimes_{\mathbb{Z}_p}\varprojlim_n I(\Gamma_n)^e/I(\Gamma_n)^{e+1}$$

such that $\nu_n(\kappa_n)=\mathcal{N}_n(\epsilon_{L_n/k,S,T}^V)$ for all $n$ and

$$e_\chi\kappa=(-1)^{re}e_\chi\operatorname{Rec}_\infty(\epsilon_{L/k,S,T}^{V'}).$$

This is the inverse-limit form of the finite-level congruence conjecture, encoding compatibility of Rubin–Stark elements across the Iwasawa tower.