The zip-cone conjecture for Hodge-type Shimura varieties

Let $G$ be the reductive group over $\mathbb{F}_p$ attached to a Hodge-type Shimura variety, let $\mu$ be its Hodge cocharacter, and let $\mathop{\mathrm{Zip}}\nolimits_G^\mu$ be the associated stack of $G$-zips. Let $C_K(\overline{\mathbb{F}}_p)$ be the cone of weights with nonzero automorphic forms and let $\mathcal C(\overline{\mathbb{F}}_p)$ be its saturation. Let $C_{\mathrm{zip}}$ be the cone of characters $\lambda$ for which $H^0(\mathop{\mathrm{Zip}}\nolimits_G^\mu,\mathcal V_I(\lambda))\neq 0$, and let $\mathcal C_{\mathrm{zip}}$ be its saturation. The zip-cone conjecture. For any Hodge-type Shimura variety, one has

$$\mathcal C(\overline{\mathbb{F}}_p)=\mathcal C_{\mathrm{zip}}.$$

This predicts that the saturated cone of weights of automorphic forms in characteristic $p$ is determined entirely by the stack of $G$-zips. It is proved in several low-dimensional cases, including Hilbert--Blumenthal varieties, Picard modular surfaces at split primes, Siegel threefolds, and additional small-rank unitary cases, but remains open in general.