Mazur–Rubin–Sano conjecture for Rubin–Stark elements

Let $\mathfrak{n}$ be a product of primes in the indexing set, let $\mathcal{H}_\mathfrak{n}=\operatorname{Gal}(F(\mathfrak{n})/F)$, let $I_\mathfrak{n}$ be the augmentation ideal of $\mathbb{Z}[\mathcal{H}_\mathfrak{n}]$, and let $c_\mathfrak{n}$ and $\epsilon_\mathfrak{n}$ be the $(p,\chi)$-components of the Rubin–Stark elements defined for $LF(\mathfrak{n})/K$ and $LF/K$, respectively. Let

$$\iota_\mathfrak{n}:{\bigcap}_{\mathcal{O}[\Gamma]}^rU_{F,S_\mathfrak{n}}\otimes_\mathbb{Z}I_\mathfrak{n}^{\nu(\mathfrak{n})}/I_\mathfrak{n}^{\nu(\mathfrak{n})+1}\hookrightarrow {\bigcap}_{\mathcal{O}[\mathcal{G}_\mathfrak{n}]}^rU_{F(\mathfrak{n}),S_\mathfrak{n}}\otimes_\mathbb{Z}\mathbb{Z}[\mathcal{H}_\mathfrak{n}]/I_\mathfrak{n}^{\nu(\mathfrak{n})+1}$$

and let ${\rm Rec}_\mathfrak{n}$ be the reciprocity homomorphism.

Mazur–Rubin–Sano conjecture. In the module

$${\bigcap}_{\mathcal{O}[\mathcal{G}_\mathfrak{n}]}^rU_{F(\mathfrak{n}),S_\mathfrak{n}}\otimes_\mathbb{Z}\mathbb{Z}[\mathcal{H}_\mathfrak{n}]/I_\mathfrak{n}^{\nu(\mathfrak{n})+1}$$

one has

$$\sum_{\sigma\in\mathcal{H}_\mathfrak{n}}\sigma c_\mathfrak{n}\otimes\sigma^{-1}=\iota_\mathfrak{n}\left({\rm Rec}_\mathfrak{n}(\epsilon_\mathfrak{n})\right).$$

This predicts a refined congruence between Rubin–Stark elements in ray-class towers, relating augmentation-ideal leading terms to local reciprocity. The supplied text does not state a proof or disproof.