The equivariant Beilinson conjecture for abelian extensions

Let $L/K$ be an abelian extension of number fields, let $G=\operatorname{Gal}(L/K)$, and let $k$ be the relevant integer. The finite motivic cohomology is $H^1_{\mathscr M}(\operatorname{Spec}L,\mathbb Q(k))_{\mathrm{fin}}$, the real Deligne cohomology is $H^1_{\mathscr D}((\operatorname{Spec}L)_{/\mathbb R},\mathbb R(k))$, and $B_L^+$ is the invariant Betti--de Rham lattice in the latter. Under the $\mathbb R[G]$-isomorphism

$$\operatorname{det}_{\mathbb R[G]}(\mathbb R\otimes_{\mathbb Q}H^1_{\mathscr M}(\operatorname{Spec}L,\mathbb Q(k))_{\mathrm{fin}})\cong\operatorname{det}_{\mathbb R[G]}(H^1_{\mathscr D}((\operatorname{Spec}L)_{/\mathbb R},\mathbb R(k))),$$

Equivariant Beilinson conjecture. The equality

$$\operatorname{det}_{\mathbb Q[G]}(H^1_{\mathscr M}(\operatorname{Spec}L,\mathbb Q(k))_{\mathrm{fin}})=L^*(1-k)\bigl(\operatorname{det}_{\mathbb Q[G]}B_L^+\bigr)$$

holds, where $L^*(1-k)\in\mathbb R[G]$ maps to $(L^*(\chi^{-1},1-k))_\chi$ under the character decomposition of $\mathbb R[G]$.

This is presented as part of the equivariant version of Beilinson's conjectures. The supplied text gives no resolution, so the conjecture remains open here.