Dahlquist–Essouabri conjecture on Euler products of polynomial factors

Let $W(x,y)=\sum_{n,m}a_{n,m}x^ny^m$ be an integral polynomial with $W(x,0)=1$, and define

$$D(s)=\prod_p W(p,p^{-s}).$$

If $\beta=\max\{n/m:a_{n,m}\neq 0\}$, then the Euler-product continuation conjecture. $D(s)$ is meromorphically continuable to the whole complex plane if and only if it is a finite product of Riemann zeta-functions; moreover, in that case the line $\Re s=\beta$ is the natural boundary of $D$. The conjecture concerns the analytic continuation and natural boundaries of Euler products built from polynomial local factors.