Beshenov's special-value conjecture for Weil-étale cohomology

Let $X$ be an arithmetic scheme and let $n<0$ be a strictly negative integer. Assume $\mathbf{L}^c(X_{\text{ét}},n)$, $\mathbf{B}(X,n)$, and the meromorphic continuation of $\zeta(X,s)$ around $s=n$. Let $\lambda$ be the canonical isomorphism

$$\lambda\colon\mathbb{R}\xrightarrow{\cong}\left(\operatorname{det}_{\mathbb{Z}}R\Gamma_{\mathrm{W,c}}(X,\mathbb{Z}(n))\right)\otimes\mathbb{R}.$$

Beshenov's special-value conjecture. The special value $\zeta^*(X,n)$ is determined up to sign by

$$\lambda\bigl(\zeta^*(X,n)^{-1}\bigr)\cdot\mathbb{Z}=\operatorname{det}_{\mathbb{Z}}R\Gamma_{\mathrm{W,c}}(X,\mathbb{Z}(n)).$$

This is the determinant-line formulation of the special-value conjecture and extends the corresponding result of Flach and Morin from proper regular arithmetic schemes to the setting considered here. It remains conditional on the finiteness, regulator, and meromorphic-continuation assumptions.