Derived $p$-adic leading term conjecture

Let $\mathcal O$ be the coefficient ring, let $I$ be the relevant augmentation ideal, let $c\in\bigcap_{\Lambda_I}^r H^1_\Sigma(\mathcal O_{K,S},\mathbb T)_I$ be the special element, let $\eta$ be the extended special element attached to $T$, let $R^{(k_0)}$ be the $k_0$-th derived Bockstein regulator, and let

$$\mathcal D:I^\varrho\cdot\bigcap_{\Lambda_I}^rH^1_\Sigma(\mathcal O_{K,S},\mathbb T)_I\longrightarrow \mathbb Q_p\otimes_{\mathbb Z_p}\bigwedge_{\mathcal O}^rH^1_\Sigma(\mathcal O_{K,S},T)\otimes_{\mathcal O}Q^\varrho$$

be the descent map. Derived $p$-adic leading term conjecture. One has

$$c\in I^\varrho\cdot\bigcap_{\Lambda_I}^rH^1_\Sigma(\mathcal O_{K,S},\mathbb T)_I,$$

and, if this holds,

$$\mathcal D(c)=R^{(k_0)}(\eta)$$

in $\mathbb C_p\otimes_{\mathbb Z_p}\bigwedge_{\mathcal O}^rH^1_\Sigma(\mathcal O_{K,S},T)\otimes_{\mathcal O}Q^\varrho$. This refines the leading-term conjecture of Kataoka–Sano; the source notes that the equality is nontrivial for higher derived Bockstein order and does not prove the conjecture in general.