Bhargava–Malle conjecture for marked squarefree extensions over function fields

Fix $n\geq 1$. Let $N(k)$ be the number of isomorphism classes of marked degree-$n$ extensions of $\mathbb{F}_q(t)$ with squarefree discriminant of degree $k$, totally split above $\infty$, and an ordering of the places above $\infty$. Let $D(k)$ be the number of squarefree monic polynomials of degree $k$. Bhargava–Malle conjecture. For even integers $k$,

$$\lim_{k\to\infty}\frac{N(k)}{D(k)}=1.$$

This is a squarefree-discriminant function-field variant of conjectures of Bhargava and Malle; the paper relates the asserted asymptotic to homological vanishing for Hurwitz spaces.