Kurokawa's absolute Euler product conjecture for arithmetic schemes

Let $X$ be an arithmetic scheme. Its counting function $N(t)$ satisfies $N(1)=\chi_{\mathrm{abs}}(X)$, the absolute Euler characteristic. An absolute Euler product is an infinite product expansion in terms of integers $\kappa(n,X)$, one for each $n\in\mathbb{N}$. Kurokawa's absolute Euler product conjecture. There should exist integers $\kappa(n,X)\in\mathbb{Z}$ such that

$$\zeta^{\mathrm{abs}}_{X}(s)=\left(\frac{1}{s}\right)^{\chi_{\mathrm{abs}}(X)}\prod_{n=1}^{\infty}\left(1-\left(\frac{1}{s}\right)^n\right)^{-\kappa(n,X)},$$

and this infinite product should converge absolutely for $\operatorname{Re}(s)>\dim X$. Kurokawa proposed this as an analogue of the Euler product for congruent zeta functions and as a representation of the absolute zeta function, whose general definition was not yet available in the stated setting; the existence of such a product and its convergence remain open here.