Hodge-Chern classes and strata-effectivity in tautological rings

Given a connected, reductive Fₚ-group G, a cocharacter μ ∈ X_*(G) and a smooth zip period map ζ:X →G- Zipμ, we study which classes in the Wedhorn-Ziegler tautological rings T^*(X), T^*(Y) of X and its flag space Y → G-ZipFlagμ are strata-effective, meaning that they are non-negative rational linear combinations of pullbacks of classes of zip (flag) strata closures. Two special cases are: (1) When X=G-Zipμ and the tautological rings ^*(X)=CHQ(G-Zipμ), T^*(Y)=CHQ(G-ZipFlagμ) are the entire Chow ring, and (2) When X is the special fiber of an integral canonical model of a Hodge-type Shimura variety -- in this case the strata are also known as Ekedahl-Oort strata. We focus on the strata-effectivity of three types of classes: (a) Effective tautological classes, (b) Chern classes of Griffiths-Hodge bundles and (c) Generically w-ordinary curves. We connect the question of strata-effectivity in (a) to the global section `Cone Conjecture' of Goldring-Koskivirta. For every representation r of G, we conjecture that the Chern classes of the Griffiths-Hodge bundle associated to (G, μ,r) are all strata-effective. This provides a vast generalization of a result of Ekedahl-van der Geer that the Chern classes of the Hodge vector bundle on the moduli space of principally polarized abelian varieties g,Fₚ in characteristic p are represented by the closures of p-rank strata. We prove several instances of our conjecture