The absolute-convergence conjecture for the auxiliary $Z$-series

Let $a_2\in\{1,\theta_0\}$, let $\mathbf s=(s_1,\ldots,s_{r+1})$, and let $\mu$ be the Möbius function on monic polynomials in $\mathbb F_q[x]$. Let $\mathrm D^{\mathrm{re}}$ be the real-root product defined in the source.

Absolute-convergence conjecture. The series

$$\sum_{h\ \mathrm{monic}}\mu(h)Z(\mathbf s,\chi_{a_2};h)$$

is absolutely convergent when $\Re(s_i)\ge\frac12$ for $i=1,\ldots,r$ and $\Re(s_{r+1})>\frac12$, away from the zero set of

$$\mathrm D^{\mathrm{re}}(\mathbf s-\mathbf{\tfrac12}).$$

This is presented as the additional analytic input which, together with the preceding holomorphy conjecture, would imply meromorphic continuation of the relevant generating function. Its resolution is not supplied in the source.