Bhargava–Cremona–Fisher–Gajović density functional-equation conjecture

Let $n$ be a positive integer and let $\sigma$ be a splitting type of degree $n$. For a prime $p$, let $\rho(n,\sigma;p)$ be the Haar measure of the $p$-adic polynomials of degree at most $n$ with splitting type $\sigma$, let $\alpha(n,\sigma;p)$ be the measure of those that are monic, and let $\beta(n,\sigma;p)$ be the measure of those that are monic and congruent to $x^n$ modulo $p$. Bhargava–Cremona–Fisher–Gajović conjecture. The densities $\rho(n,\sigma;p)$, $\alpha(n,\sigma;p)$, and $\beta(n,\sigma;p)$ are rational functions in $p$ and satisfy

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

The paper proves rationality in the tamely ramified case and notes that the conjecture predicts rationality beyond the gcd condition on the residue characteristic and ramification indices; the full assertion is not proved there.