Explicit Drinfeld moduli schemes and Abhyankar's generalized iteration conjecture

Let $k$ be a field containing ${\mathbb{F}}_q$. Let $\psi$ be a rank $r$ Drinfeld ${\mathbb{F}}_q[t]$-module determined by $\psi_t(X) = tX+a_1X^q+\cdots+a_{r-1}X^{q^{r-1}}+X^{q^r}$,where$t,a_1,\ldots,a_{r-1}$ are algebraically independent over $k$. Let $n\in {\mathbb{F}}_q[T]$ be a monic polynomial. We show that the Galois group of ${\psi }_n(X)$ over $k(t,a_1,\dots ,a_{r-1})$ is isomorphic to ${\mathrm{G}\mathrm{L}}_r({\mathbb{F}}_q[t]\mathrm{/}n{\mathbb{F}}_q[t])$, settling a conjecture of Abhyankar. Along the way we obtain an explicit construction of Drinfeld moduli schemes of level $tn$.