Gross–Reeder conjecture for adjoint Swan conductors

Let $k$ be a $p$-adic field with residue field $\mathfrak{f}$, and let $\mathcal{W}=\langle \operatorname{Fr}\rangle\ltimes\mathcal{I}$ be its Weil group, with wild inertia subgroup $\mathcal{I}_+$. Let $G$ be a connected reductive algebraic group over $\mathbb{C}$, and let a Langlands parameter be a homomorphism

$$\phi:\mathcal{W}\times \operatorname{SL}_2(\mathbb{C})\to G$$

whose restriction to $\operatorname{SL}_2(\mathbb{C})$ is algebraic, whose restriction to $\mathcal{I}$ is continuous, and for which $\phi(\operatorname{Fr})$ is semisimple. The parameter is discrete if the centralizer of its image is finite, and inertially discrete if the action of $\phi(\mathcal{I})$ on the Lie algebra $\mathfrak{g}$ has no nonzero invariants. For the adjoint map $\operatorname{Ad}:G\to\operatorname{GL}(\mathfrak{g})$, write the Swan conductor of the resulting parameter as the Swan conductor of $\operatorname{Ad}(\phi)$. Gross–Reeder conjecture. Suppose $G$ is simple. If the Langlands parameter $\phi:\mathcal{W}\times \operatorname{SL}_2(\mathbb{C})\to G$ is discrete and inertially discrete, then the Swan conductor of $\operatorname{Ad}(\phi)$ is greater than or equal to the rank of $G$. This conjecture predicts the minimum possible wild ramification of a discrete, inertially discrete Langlands parameter; the paper presents its geometric analogue and proves the corresponding lower bound for irregular flat $G$-bundles, with equality characterized by Frenkel–Gross connections.