A geometric analogue of a conjecture of Gross and Reeder

Let $G$ be a simple complex algebraic group. We prove that the irregularity of the adjoint connection of an irregular flat $G$-bundle on the formal punctured disk is always greater than or equal to the rank of $G$. This can be considered as a geometric analogue of a conjecture of Gross and Reeder. We will also show that the irregular connections with minimum adjoint irregularity are precisely the (formal) Frenkel-Gross connections. As a corollary, we establish the de Rham analogue of a conjecture of Heinloth, Ng\^o, and Yun for $G={\rm SL}_n$.