Bhargava–Cremona–Fisher–Gajović symmetry conjecture for polynomial factorization densities

Let $h(z)$ be a degree-$n$ polynomial over a local field with residue-field size $q$, and let $\rho(n,\sigma;q)$ denote the density of polynomials whose associated finite étale algebra has factorization type $\sigma$. The density is understood with respect to normalized Haar measure, and the analogous densities $\alpha(n,\sigma;q)$ for monic integral polynomials and $\beta(n,\sigma;q)$ for polynomials congruent to $z^n$ modulo the maximal ideal are defined similarly. Bhargava–Cremona–Fisher–Gajović conjecture. The quantities $\rho(n,\sigma;q)$, $\alpha(n,\sigma;q)$ and $\beta(n,\sigma;q)$ are rational functions of $q$ satisfying

$$\rho(n,\sigma;q^{-1})=\rho(n,\sigma;q),$$

and

$$\alpha(n,\sigma;q^{-1})=\beta(n,\sigma;q).$$

Implicitly, these densities depend on the local field only through the size $q$ of its residue field. The paper proves this conjecture, generalizing the original polynomial-density statement to local fields; consequently, its database status is solved.