Beshenov's regulator conjecture for Weil-étale motivic cohomology

Let $X$ be an arithmetic scheme such that $X_{\mathbb{C}}$ is smooth and quasi-projective, and let $n<0$. Let $G_{\mathbb{R}}=\operatorname{Gal}(\mathbb{C}/\mathbb{R})$, let $\mathbb{R}(n)$ denote the real Tate twist, and let $Reg_{X,n}$ be the étale regulator. Beshenov's regulator conjecture. The induced morphism

$$Reg_{X,n}^{\vee}\colon R\Gamma_c(G_{\mathbb{R}},X(\mathbb{C}),\mathbb{R}(n))[-1]\longrightarrow R\operatorname{Hom}(R\Gamma(X_{\text{ét}},\mathbb{Z}^c(n)),\mathbb{R})$$

should be a quasi-isomorphism of complexes of real vector spaces. The regulator is intended to encode the contribution of the Archimedean places to the Weil-étale theory. The source states no resolution and notes that defining it for singular complex fibers remains future work.