The two-variable Beilinson–Flach element main conjecture

Let $\mathcal{K}$ be the imaginary quadratic field in the source, let $\Lambda_\mathcal{K}$ be its two-variable Iwasawa algebra, let $H^1_3(\mathcal{K},M(f)\otimes\Lambda_\mathcal{K})$ be the indicated Selmer group, and let $BF^+$ be the Beilinson–Flach element. Let $X_{v_0,+}$ be the Pontryagin dual of the corresponding Selmer group. The two-variable Beilinson–Flach length conjecture. For every height-one prime $P$ of $\Lambda_\mathcal{K}$, the localization at $\Lambda_P$ of the quotient

$$H^1_3(\mathcal{K},M(f)\otimes\Lambda_\mathcal{K})/\Lambda_\mathcal{K}\cdot BF^+$$

has the same length as the localization of $X_{v_0,+}$. The source presents this as a version of the preceding two-variable main conjecture and notes that the Selmer group is torsion-free of rank one; the paper proves a weak version and one divisibility rather than this full equality.