Prescribed-coefficients point-count formula for irreducible polynomials over finite fields

Let $q=p^r$, let $1\le l_0<\cdots<l_{m-1}$, let $(t_{l_0},\ldots,t_{l_{m-1}})\in(\mathbb{F}_q)^m$, and let $\overline{n}\in\\{0,\ldots,p^{1+\lfloor\log_p l_{m-1}\rfloor}-1\\}$. Write $F_q(n,t_{l_0},\ldots,t_{l_{m-1}})$ for the number of irreducible polynomials of degree $n$ over $\mathbb{F}_q$ with the prescribed coefficients indexed by $l_0,\ldots,l_{m-1}$.

Prescribed-coefficients enumeration conjecture. There exist $\omega_1,\ldots,\omega_N\in\overline{\mathbb{Z}}$, all of norm $\sqrt q$, $\upsilon_1,\ldots,\upsilon_N\in\mathbb{Z}$, and an integer $s\ge0$ such that, for every $n\ge l_{m-1}$ satisfying

$$n\equiv\overline{n}\pmod{p^{1+\lfloor\log_p l_{m-1}\rfloor}},$$

one has

$$F_q(n,t_{l_0},\ldots,t_{l_{m-1}})=\frac{1}{q^m}\left(q^n+\frac{1}{q^s}\sum_{i=1}^N\upsilon_i\alpha_i^n\right)=q^{n-m}+O(q^{n/2}).$$

Here the claim is understood uniformly for every choice of the displayed finite-field data and residue class.