Strata-effectivity conjecture for Griffiths-Hodge Chern classes

Let $G$ be a connected reductive group over $\boldsymbol{F}_p$, let $\mu$ be a cocharacter of $G$, and let $r$ be a representation of $G$. The associated Griffiths-Hodge bundle is denoted by \operatorname{Grif}(\mathop{\text{G-\tt Zip}}\nolimits^{\mu},r). Strata-effectivity conjecture. For every triple $(G,\mu,r)$, all the Chern classes of \operatorname{Grif}(\mathop{\text{G-\tt Zip}}\nolimits^{\mu},r) are strata-effective. This generalizes the known strata-effectivity of the Chern classes of the Hodge vector bundle in the Siegel case, and would also imply corresponding statements after pullback along smooth surjective zip period maps.