Integrality Conjecture for Bloch–Kato elements

Let $M/K$ be a rank-one motive with coefficient ring $R$, let $T$ be its $p$-adic realization, and put $r=r(M/K,R)$. Assume that the $\mathcal{R}_p$-module $Y_K(T):=\bigoplus_{v\text{ "}in S_\infty(K)}H^0(K_v,T)$ is free. Let $S$ contain $S_\infty(K)\cup S_p(K)\cup S_{\rm ram}(T)$, let $b$ be an ordered $\mathcal{R}_p$-basis of $Y_K(T)$, and let $\eta_S^b(T)$ be the associated Bloch–Kato element. Assume also that $H^0(K,T^\ast(1))=0$ and that $H^1(\mathcal{O}_{K,S},T^\ast(1))$ is $\mathds{Z}_p$-free. Integrality Conjecture. Under these hypotheses,

$$\eta_S^b(T)\in {\bigcap}_{\mathcal{R}_p}^rH^1(\mathcal{O}_{K,S},T^\ast(1)).$$

This conjecture asserts the expected integral refinement of the Bloch–Kato element, replacing its a priori $\mathds{C}_p$-valued exterior-power realization by the exterior-power bidual lattice. Its general validity is not established in the source.