  For a connected, reductive group $G$ over a finite field endowed with a
cocharacter $\mu$, we define the zip cone of $(G,\mu)$ as the cone of all
possible weights of mod $p$ automorphic forms on the stack of $G$-zips. This
cone is conjectured to coincide with the cone of weights of characteristic $p$
automorphic forms for Hodge-type Shimura varieties of good reduction. We prove
in full generality that the cone of weights of characteristic $0$ automorphic
forms is contained in the zip cone, which gives further evidence to this
conjecture. Furthermore, we determine exactly when the zip cone is generated by
the weights of partial Hasse invariants, which is a group-theoretical
generalization of a result of Diamond--Kassaei and Goldring--Koskivirta.
