The filtration–singular-support conjecture for nilpotent subschemes

Let $G$ be a reductive group with Lie algebra $\mathfrak{g}$, let $X$ be the underlying curve, and let $Y\tsubset {\mathcal N}(\mathfrak{g})$ be a closed $G$-invariant subscheme. Let $F_Y$ be the full subcategory of $Shv(\operatorname{Bun}_G)$ consisting of objects with no Fourier coefficients corresponding to nilpotent orbits not contained in $Y$, and let $F_{\operatorname{Nilp},Y}=F_Y\cap Shv_{\operatorname{Nilp}}(\operatorname{Bun}_G)$. Let $Shv_Y(\operatorname{Bun}_G)$ be the full subcategory of objects whose singular support is contained in the corresponding mapping stack defined by $Y$.

The filtration–singular-support conjecture.

$$F_{\operatorname{Nilp},Y}=Shv_Y(\operatorname{Bun}_G).$$

This conjecture asserts that, within the nilpotent-singular-support category, the Fourier-coefficient filtration is exactly the filtration by singular support. The source provides no resolution status.