  A general conjecture is stated on the cone of automorphic vector bundles
admitting nonzero global sections on schemes endowed with a smooth, surjective
morphism to a stack of $G$-zips of connected-Hodge-type; such schemes should
include all Hodge-type Shimura varieties with hyperspecial level. We prove our
conjecture for groups of type $A_1^n$, $C_2$ and $\mathbf F_p$-split groups of
type $A_2$ (this includes all Hilbert-Blumenthal varieties and should also
apply to Siegel modular threefolds and Picard modular surfaces). An example is
given to show that our conjecture can fail for zip data not of
connected-Hodge-type.
