Rubin-type main conjecture for supersingular Iwasawa modules

Let $F_f$ be a height $2$ and self-dual formal group, and let $\mathbb{U}'$ and $\mathbb{X}'$ be the torsion $\Lambda(G_{\infty},\mathcal{O}_{L_p})$-modules occurring in the paper. For $\chi\in\hat{\Delta}'$, write $\mathbb{U}'_{\chi}$ and $\mathbb{X}'_{\chi}$ for their $\chi$-components, and let $\mu_{\mathrm{glob}}(\frak{g};\chi)$ be the specialization of the global measure at $\chi$. Rubin-type main conjecture. One has

$$\operatorname{det}_{\Lambda(G_{\infty},\mathcal{O}_{L_p})}(\mathbb{U}')=\operatorname{det}_{\Lambda(G_{\infty},\mathcal{O}_{L_p})}(\mathbb{X}').$$

Moreover, for every nontrivial $\chi\in\hat{\Delta}'$,

$$\mu_{\mathrm{glob}}(\frak{g};\chi)\Lambda(\Gamma',\mathcal{O}_{L_p,\chi})=\operatorname{char}_{\Lambda(\Gamma',\mathcal{O}_{L_p,\chi})}(\mathbb{U}'_{\chi})=\operatorname{char}_{\Lambda(\Gamma',\mathcal{O}_{L_p,\chi})}(\mathbb{X}'_{\chi}).$$

This is a Rubin-type equality between global Euler-system data and Iwasawa-theoretic class-group data. The formulation for the trivial character requires a suitable modification accounting for a pole; the supplied text does not establish the conjecture in general.