Galois groups associated to generic Drinfeld modules and a conjecture of Abhyankar

Let $\varphi$ be a rank $r$ Drinfeld ${\text{BF}}_q[T]$-module determined by $\phi_T(X) = TX+g_1X^q+…+g_{r-1}X^{q^{r-1}}+X^{q^r}$,where$g_1,…,g_{r-1}$ are algebraically independent over ${\text{BF}}_q(T)$. Let $N\in {\text{BF}}_q[T]$ be a polynomial, and $k\mathrm{/}{\text{BF}}_q$ an algebraic extension. We show that the Galois group of ${\varphi }_N(X)$ over $k(T,g_1,\mathrm{.}\mathrm{.}\mathrm{.},g_{r-1})$ is isomorphic to ${\text{GL}}_r({\text{BF}}_q[T]\mathrm{/}N{\text{BF}}_q[T])$, settling a conjecture of Abhyankar.