Perrin-Riou's nondegeneracy conjecture for the dihedral $p$-adic height pairing

Let $A$ be an abelian variety over $F$ in the dihedral setting above $K$, let $G=\operatorname{Gal}(K_{p^\infty}/K)$, and write $\Lambda=\Lambda(G)={\bf Z}_p[[G]]$. Let $\mathfrak h_\infty$ be Perrin-Riou's $\Lambda$-adic height pairing, let $\mathcal R$ be the associated $p$-adic regulator, let $\mathfrak K$ be the inverse limit of the relevant Artin symbols, and let $\gamma$ be a topological generator of the cyclotomic quotient $\Gamma$. Perrin-Riou's regulator conjecture. (i) The $G$-Euler characteristic $\chi(G,\operatorname{Sel}_{p^\infty}(A/K_{p^\infty}))$ is well defined; equivalently, the image in $\Lambda$ of $\mathcal L_f=\mathcal L_{f,1}(\gamma-1)$ is not identically zero, or equivalently, $\mathfrak h_\infty$ is nondegenerate. (ii) The regulator $\mathcal R$ is nondegenerate and

$$\mathcal R={\bf e}(\mathfrak K\cdot\log_p(\gamma))^{-1}\Lambda.$$

The conjecture connects nonvanishing of the two-variable $p$-adic $L$-function with nondegeneracy of the $p$-adic height pairing and gives an explicit regulator ideal; the supplied text does not establish it.