Arveson’s hyperrigidity conjecture concerns the unique extension property (UEP) of representations of C*-algebras with respect to a generating operator system. Bilich and Dor-On showed that the conjecture does not hold in general. Nevertheless, the conjecture remains interesting for several reasons. The states that are maximal in the dilation order fully encapsulate the cyclic representations of a C*-algebra possessing the UEP. The collection of all such maximal states forms a norm-closed set that remains stable under absolute continuity. In this talk, we will discuss an equivalent characterization of the dilation-maximal states in terms of a boundary projection. Subsequently, we will present a reformulation of Arveson’s hyperrigidity conjecture in terms of the non-commutative topological properties of this boundary projection. This is joint work with Raphaël Clouâtre.