Equivariant Tamagawa number conjecture for motives

Let $K'$ be a finite extension of $K$, let $\varepsilon$ be an idempotent of $A[\mathcal{G}_{K'}]$, and let $\mathcal{R}$ be a $\mathbb{Z}_p$-order in $A[\mathcal{G}_{K'}]\varepsilon$. Let $C_{K',S}(T)$ be the compactly supported cohomological complex, let $L^\ast(M_{K'/K}^\vee,0)$ be the leading term at $s=0$ of the equivariant $L$-function, and let

$$\vartheta_{M,K',S}^{\rm BK}:\mathbb{C}_p\otimes_{\mathbb{Z}_p}\det_{\mathcal{A}[\mathcal{G}_{K'}]}(C_{K',S}(T))\simeq(A_{\mathbb{C}_p}[\mathcal{G}_{K'}],0)$$

be the canonical isomorphism. Equivariant Tamagawa number conjecture. In $(A_{\mathbb{C}_p}[\mathcal{G}_{K'}]\varepsilon,0)$, there is an equality of graded invertible $\mathcal{R}$-modules

$$\mathcal{R}\cdot\vartheta_{M,K',S}^{\rm BK}\!\left(\det_{\mathcal{A}[\mathcal{G}_{K'}]}(C_{K',S}(T))\right)=(\mathcal{R}\cdot L^\ast(M_{K'/K}^\vee,0),0).$$

This is the equivariant Tamagawa number conjecture for $(M_{K'}^\vee,\mathcal{R})$. The paper also notes that the construction of the isomorphism can depend on motivic-cohomology conjectures in general, although it is unconditional in the principal settings considered.