The Hadamard meta-conjecture for natural sequences of zeta functions

Let $q$ be a prime power, let $a_n\in\mathcal{M}_{\mathbb{F}_q}$ be a natural sequence of classes, and let $\mathcal{H}_1$ denote the set of Hadamard functions $f$ satisfying $f(0)=1$. Suppose that the zeta functions $Z_{a_n}(t)$ converge in both the point-counting and weight topologies to some $f(t)\in\mathcal{H}_1$. Hadamard meta-conjecture. Then

$$Z_{a_n}(t)\longrightarrow f(t)$$

in the Hadamard topology.

The Hadamard topology refines the point-counting and weight topologies, while its limits retain meromorphic-function structure. The claim proposes a common refinement for natural asymptotic results in arithmetic and motivic statistics; no resolution is supplied in the source.