Coxeter or building interpretation of the Reye projection

Determine whether the map from $D_4$ root geometry to Reye geometry can be realized as a genuine typed shadow or residue construction in a building or related chamber geometry, and, if not, identify the exact obstruction.

Background

The paper observes that the 24-cell and the Weyl group $W(F_4)$ suggest a Coxeter-theoretic interpretation of the passage from the $D_4$ root configuration to the Reye configuration.

No building structure is assumed in the paper, and it remains unresolved whether the antipodal/projective reduction can be understood as a typed shadow or residue construction, or whether a precise obstruction prevents such an interpretation.