The cyclotomic and anticyclotomic main conjectures

Let $L_{\mathrm{cycl}}$ be the cyclotomic specialization of the two-variable $p$-adic $L$-function, let $L'$ be its linear cyclotomic term over the anticyclotomic line, and let $\Selp{K_\infty^{\mathrm{anti}}}_{\mathrm{tors}}$ denote the torsion submodule of the anticyclotomic Selmer module.

Cyclotomic and anticyclotomic main conjectures. Both assertions hold:

1. $L_{\mathrm{cycl}}$ generates $\operatorname{char}_{\Lambda_{\mathrm{cycl}}}(\Selp{K_\infty^{\mathrm{cycl}}})$ in $\Lambda_{\mathrm{cycl}}$.
2. $L'$ generates

$$\Gamma_{\mathrm{cycl}}\otimes\operatorname{char}_{\Lambda_{\mathrm{anti}}}(\Selp{K_\infty^{\mathrm{anti}}}_{\mathrm{tors}})$$

inside $\Gamma_{\mathrm{cycl}}\otimes\Lambda_{\mathrm{anti}}$.

These are the one-variable specializations of the two-variable main conjecture. The source records divisibilities related to both assertions but does not state equality is known.