Beshenov's vanishing-order conjecture for arithmetic schemes

Let $X$ be an arithmetic scheme and let $n<0$ be an integer. Assume that $\zeta(X,s)$ has a meromorphic continuation around $s=n$, and let $H^i_{\mathrm{W,c}}(X,\mathbb{Z}(n))$ denote the Weil-étale cohomology groups with compact support. Beshenov's vanishing-order conjecture. One has

$$\operatorname{ord}_{s=n}\zeta(X,s)=\sum_{i\in\mathbb{Z}}(-1)^i\,i\,\operatorname{rk}_{\mathbb{Z}}H^i_{\mathrm{W,c}}(X,\mathbb{Z}(n)).$$

This gives the order of the zero or pole of the zeta function in terms of Weil-étale cohomology. The paper verifies compatibility with several geometric decompositions and obtains cases such as cellular schemes over suitable one-dimensional bases, but does not establish the conjecture in general.