The generalized Perrin-Riou conjecture for finite-level Darmon norms

Assume the paper's Hypothesis and set $r=r_{\rm alg}$. Let $F/\mathbb Q$ be the fixed finite real abelian extension, let $I_F$ be the augmentation ideal, let ${}_{c,d}z_F$ be the modified Kato zeta element, let ${}_{c,d}\eta_{\bm{x}}^{\rm BSD}$ be the modified Birch–Swinnerton-Dyer element, let ${\rm Boc}_{F,\bm{x}}$ be the Bockstein regulator, and let $\iota_F$ be the map induced by restriction and the inclusion $Q_F^{r-1}\hookrightarrow\mathbb Z_p[G]/I_F^r$. Generalized Perrin-Riou conjecture. For every $\mathbb Z_p$-basis element $\bm{x}$ of $\bigwedge_{\mathbb Z_p}^{r-1}H^2(\mathbb Z_S,T)_{\rm tf}$, ${}_{c,d}\eta_{\bm{x}}^{\rm BSD}$ belongs to $\bigwedge_{\mathbb Z_p}^rH^1(\mathbb Z_S,T)$, and the image of the Darmon norm $\mathcal N_{F/\mathbb Q}({}_{c,d}z_F)$ in $H^1(\mathcal O_{F,S},T)\otimes_{\mathbb Z_p}\mathbb Z_p[G]/I_F^r$ equals $\iota_F({\rm Boc}_{F,\bm{x}}({}_{c,d}\eta_{\bm{x}}^{\rm BSD}))$. This is the precise finite-level formulation of the generalized conjecture, refining the rank-zero and rank-one Perrin-Riou prediction; it remains open in general.