Anticyclotomic Iwasawa main conjecture for the BDP $p$-adic $L$-function

Let $K$ be an imaginary quadratic field satisfying the Heegner, discriminant, and splitting hypotheses, let $K_\infty$ be its anticyclotomic $Z_p$-extension, let $\Lambda$ be the associated Iwasawa algebra, and let $\mathfrak{X}_{\rm Gr}(E/K_\infty)$ be the Pontryagin dual of the Greenberg Selmer group. Let $\Lambda^{\rm ur}$ denote the relevant unramified extension of $\Lambda$, and let $\mathcal{L}_p^{\rm BDP}(f/K)$ be the BDP $p$-adic $L$-function. Anticyclotomic main conjecture. The module $\mathfrak{X}_{\rm Gr}(E/K_\infty)$ is $\Lambda$-torsion, and

$${\rm char}_\Lambda(\mathfrak{X}_{\rm Gr}(E/K_\infty))\Lambda^{\rm ur}=(\mathcal{L}_p^{\rm BDP}(f/K)^2)$$

as ideals in $\Lambda^{\rm ur}$. This is the anticyclotomic Iwasawa main conjecture for the square of the BDP $p$-adic $L$-function and is presented as a special case of Greenberg's main-conjecture framework; its general status is not resolved in the supplied text.