Moduli-theoretic interpretation for exceptional (non-abelian type) Shimura varieties

Determine a moduli-theoretic interpretation for Shimura varieties whose associated group G is not of abelian type, including all exceptional Shimura varieties; specifically, construct an explicit moduli space (e.g., of motives with appropriate G^c-structure) that these Shimura varieties represent.

Background

The authors explain that Shimura varieties are expected to be moduli spaces of motives with G-structure, and that this picture is realized in the abelian type case via motives from abelian varieties with Hodge cycles. However, they explicitly note that for a large class of Shimura varieties—including all exceptional ones—no moduli-theoretic interpretation is currently known.

Clarifying such a moduli interpretation would bridge the motivic expectations with concrete moduli spaces and would significantly advance the understanding of non-abelian type Shimura varieties.