Perrin-Riou's conjecture on the Beilinson–Kato class

Let $E$ be the elliptic curve under consideration, let $T^*$ be the relevant dual $p$-adic Galois representation, and let

$$\frak{z}_0^{\textup{BK}}\in H^1(\mathbb{Q},T^*)\otimes\mathbb{Q}_p.$$

Perrin-Riou's conjecture. The element $\frak{z}_0^{\textup{BK}}$ is non-trivial if and only if

$$\operatorname{ord}_{s=1}L(E,s)\leq 1.$$

This conjecture relates the non-vanishing of Kato's Beilinson–Kato class to the analytic rank of $E$. The source presents it as a conjecture of Perrin-Riou and does not state that it has been resolved.