Heegner Point Main Conjecture for potentially ordinary elliptic curves at Eisenstein primes

Let $K$ be the imaginary quadratic field, let $(\tilde{f},\chi_\varepsilon)$ be a Heegner pair, and let $\Lambda$ be the relevant Iwasawa algebra. Let $\bT_\varepsilon$ and $M_\varepsilon$ be the associated Galois representations, let

$$\cX={\text{\textup{H}}}^1_{\cF_{\Lambda,\varepsilon}}(K,M_\varepsilon)^\vee,$$

and let $\kappa_1\in {\text{\textup{H}}}^1_{\cF_{\Lambda,\varepsilon}}(K,\bT_\varepsilon)$ be the class arising from a Heegner point associated to $(\tilde{f},\chi_\varepsilon)$. Heegner Point Main Conjecture. The module ${\text{\textup{H}}}^1_{\cF_{\Lambda,\varepsilon}}(K,\bT_\varepsilon)$ has $\Lambda$-rank one, and there is a finitely generated torsion $\Lambda$-module $M$ such that

$$\cX\sim\Lambda\oplus M\oplus M$$

and

$$\operatorname{char}_\Lambda(M)=\operatorname{char}_\Lambda\left({\text{\textup{H}}}^1_{\cF_{\Lambda,\varepsilon}}(K,\bT_\varepsilon)/\Lambda\kappa_1\right).$$

This conjecture relates the structure of the dual ordinary Selmer group to the quotient of the rank-one Selmer module by the Heegner-point Euler-system class; the supplied text does not state whether it has been proved or remains open.