Cooper's conjectural description of tautological rings of Shimura varieties

Let $G$ be a reductive group over a field $k$, let $P$ be a parabolic subgroup, and let $\Delta$ be the set of simple roots. For each $J\subset\Delta$, let $L_J$ be the corresponding Levi subgroup, and let $[L_J/L_J\cap P]$ denote the associated class in $A^{\bullet}(G/P)$. Let $S$ be the special fibre of an integral canonical model of a Shimura variety of Hodge type. Define

$$\mathcal{J} \coloneqq \{\Delta\setminus\{\alpha\} \mid \alpha\in\Delta \text{ and }P_J\subset G \text{ defined over }\mathbb{Q}\}.$$

The set $\mathcal{J}$ carries a partial order $\leq$ induced by the stratification of the boundary of a toroidal compactification $S^{\rm tor}$ of $S$. Cooper's conjecture. The tautological ring of $S$ should satisfy

$$T^{\bullet}(S) \cong A^{\bullet}(G/P)/([L_J/L_J\cap P] \mid J\in\mathcal{J}\text{ maximal with respect to }\leq).$$

This conjecture proposes a description of the tautological ring in terms of the Chow ring of the flag variety $G/P$, extending the relationship between automorphic vector bundles and homogeneous vector bundles. Its status is not resolved by the supplied source material.