The Mertens-type conjecture for proper smooth schemes over the integers

Let $X$ be a proper smooth scheme over $\mathbf Z$, let $N(x)$ denote the norm associated with a closed point $x$ of $X$, and let $\zeta(X,s)$ be its zeta function. Write $\dim(X)$ for the dimension of $X$ and $\gamma$ for Euler's constant.

Mertens-type conjecture for schemes over $\mathbf Z$.

$$\prod_{N(x)\le t}(1-N(x)^{-\dim(X)})^{-1} \sim\left(\operatorname{Res}_{s=\dim(X)}\zeta(X,s)\right)e^\gamma\log t$$

as $t\to\infty$.

The source proves this asymptotic for proper smooth schemes over finite fields and conjectures it for general proper smooth schemes over $\mathbf Z$.