Kato's weak Tamagawa number conjecture for constructible K-theory elements

Let $H_{{\cal M}}$ be the relevant Adams eigenspace of Quillen K-theory, and let $H^{\operatorname{constr}}_{{\cal M}}\subset H_{{\cal M}}$ be the subspace of constructible elements. Retain the notation $r_{{\cal D}}$, $r_p$, $H^2_p$, $L_S(V_p^*(1),s)$, $H_{h,\mathbb{Z}}$, $e$, and the determinant lattices from Kato's Tamagawa number conjecture.

Kato's weak Tamagawa number conjecture. There is a subspace $H^{\operatorname{constr}}_{{\cal M}}$ such that $r_{{\cal D}}$ and $r_p$ restricted to it are isomorphisms and $H^2_p$ is finite; the dimension assertion is the same as part (b) of the full conjecture; there is an element $\xi\in\det_{\mathbb{Q}}(H^{\operatorname{constr}}_{{\cal M}})$ with

$$r_{{\cal D}}(\xi)=\left(\lim_{s\to0}s^{-e}L_S(V_p^*(1),s)\right)\eta;$$

and $r_p(\xi)$ is a basis of

$$\det_{\mathbb{Z}_p}(R\Gamma(\mathcal{O}_S,T_p))^{-1}\subset\det_{\mathbb{Q}_p}(R\Gamma(\mathcal{O}_S,V_p)[-1])\cong\det_{\mathbb{Q}_p}(H^1_p\otimes_{\mathbb{Z}_p}\mathbb{Q}_p).$$

The weak form replaces the full motivic K-theory space by constructible elements, while retaining the regulator, special-value, and determinant-lattice predictions.