Natural-boundary conjecture for non-cyclotomic Euler products

Let

$$h(X_1,X_2)=1+\sum_{i=1}^{r}(a_{i0}+a_{i1}X_1+\cdots+a_{in_i}X_1^{n_i})X_2^i\in\mathbb{Z}[X_1,X_2]$$

be a polynomial that is not a finite product of cyclotomic polynomials as in the Rudnick–du Sautoy conjecture, and suppose that all cyclotomic factors of $h(X_1,X_2)$ have been removed. Set

$$\beta=\operatorname{max}\left\{\frac{n_i}{i}:i\in\\{1,\dots,r\\}\right\}.$$

Natural-boundary conjecture. The function

$$Z(s)=\prod_{p\\ \ \textrm{prime}}h(p^{-s},p)$$

admits $\operatorname{Re}(s)=\beta$ as a natural boundary of meromorphy: it can be meromorphically continued to $\\{s\in\mathbb{C}:\operatorname{Re}(s)>\beta\\}$, but it has no meromorphic continuation beyond the line $\operatorname{Re}(s)=\beta$. The claim specifies the expected maximal domain for the non-cyclotomic case after cyclotomic factors are removed. Its resolution status is not supplied in the provided text.