The Iwasawa-theoretic generalized Perrin-Riou conjecture

Assume the paper's Hypothesis, set $r=r_{\rm alg}$, and let $\bm{x}$ be a $\mathbb Z_p$-basis of $\bigwedge_{\mathbb Z_p}^{r-1}H^2(\mathbb Z_S,T)_{\rm tf}$. Let $\kappa_\infty$ be the normalized Iwasawa–Darmon derivative, let $Q^{r-1}$ be the inverse-limit augmentation quotient, let $\eta_{\bm{x}}^{\rm BSD}$ be the Birch–Swinnerton-Dyer element, and let ${\rm Boc}_{\infty,\bm{x}}$ be the induced Bockstein regulator map. Iwasawa-theoretic generalized Perrin-Riou conjecture. One has

$$\kappa_\infty={\rm Boc}_{\infty,\bm{x}}(\eta_{\bm{x}}^{\rm BSD})$$

in $\mathbb C_p\otimes_{\mathbb Z_p}H^1(\mathbb Z_S,T)\otimes_{\mathbb Z_p}Q^{r-1}$. This is the infinite-level refinement of the finite-level generalized conjecture; it is equivalent to the corresponding leading-term prediction and remains open in general.