Bloch–Kato–Kato equality for elliptic curves

Let $E/\mathbb{Q}$ be the elliptic curve and let $F/\mathbb{Q}$ be the fixed totally real extension of odd degree. Assume that $\Sha(E/F)[p^\infty]$ is finite. Let $T$ be the associated $p$-adic representation, let $\underline{b}$ be the basis of $Y_F(T)$ obtained from the chosen generator of $H_1(E(\mathbb{C}),\mathbb{Z})^+$, let $\eta_F^{\rm BSD}=\eta_{\underline{b},F}^{\rm BK}$ be the Bloch–Kato element, and let $z_F^{\rm Kato}$ be Kato's normalized zeta element. Bloch–Kato–Kato conjecture.

$$\eta_F^{\rm BSD}=z_F^{\rm Kato}.$$

This conjecture compares the Bloch–Kato element, defined using the equivariant leading term and regulator data, with Kato's zeta element in $H^1(\mathcal{O}_{F,S},V)$. No resolution is stated.