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For several Hodge-type Shimura varieties of good reduction in characteristic$p$, we show that the cone of weights of automorphic forms is encoded by thestack of $G$-zips of Pink-Wedhorn-Ziegler. This establishes several instancesof a general conjecture formulated in previous papers by the authors.Furthermore, we prove in these cases that any mod $p$ automorphic form whoseweight lies in a specific region of the weight space is divisible by a partialHasse invariant. This generalizes to other Shimura varieties previous resultsof Diamond--Kassaei on Hilbert modular forms.

Divisibility of mod $p$ automorphic forms and the cone conjecture for certain Shimura varieties of Hodge-type