The Iwasawa-theoretic Mazur–Rubin–Sano conjecture

Let $K_\infty/k$ be the relevant $mathds{Z}_p$-extension, let $\Gamma$ be its pro-$p$ direction, let $W=V'\setminus V$ have cardinality $e$, and let $\varepsilon^V_{K_n/k,\Sigma,T}$ and $\varepsilon^{V'}_{K/k,\Sigma,T}$ be the corresponding Rubin–Stark elements. Let $\operatorname{Rec}_W$ be the wedge of the local reciprocity maps, let $I(\Gamma)$ be the augmentation ideal, and let $\nu_n$ and $\mathcal{N}_n$ be the transition and norm maps used in the source. Iwasawa-theoretic Mazur–Rubin–Sano conjecture. Assuming the $p$-part of the Rubin–Stark conjecture holds for every $K_n/k$ with $(V,\Sigma,T)$, there exists

$$\mathfrak{k}\in\left(\bigcap\nolimits^r_{\mathds{Z}_p[\mathcal{G}]}U_{K,\Sigma,T}\right)\otimes_{\mathds{Z}_p}\frac{I(\Gamma)^e}{I(\Gamma)^{e+1}}$$

with $\nu_n(\mathfrak{k}_n)=\mathcal{N}_n(\varepsilon^V_{K_n/k,\Sigma,T})$ for all $n$, and

$$\mathfrak{k}=(-1)^{re}\operatorname{Rec}_W(\varepsilon^{V'}_{K/k,\Sigma,T}),$$

where the equality is in

$$\mathds{Q}_p\left(\bigcap\nolimits^r_{\mathds{Z}_p[\mathcal{G}]}U_{K,\Sigma,T}\right)\otimes_{\mathds{Z}_p}\frac{I(\Gamma)^e}{I(\Gamma)^{e+1}}.$$

This is the Iwasawa-theoretic refinement relating norm-coherent Rubin–Stark elements to local reciprocity and Darmon derivatives. The source does not state a resolution.