The asymptotic polynomial-count conjecture for complex varieties

A complex variety $X$ is a variety over $\mathbb{C}$. It is asymptotically polynomial count when its point counts over finite fields agree with a polynomial in the field size for all sufficiently large prime powers in the relevant asymptotic sense, and it is polynomial motivic when its class in the Grothendieck ring $\mathrm{K}\mathbf{Var}_{\mathbb{C}}$ is $P(q)$ for some $P(t)\in\mathbb{Z}[t]$, called its motivic polynomial. The asymptotic polynomial-count conjecture. A complex variety is asymptotically polynomial count if and only if it is polynomial motivic, and the counting and motivic polynomials agree. The conjecture is contradicted by the paper's later $\mathrm{AGL}_{1}$ representation-variety example, which is polynomial motivic but not asymptotically polynomial count.