  We study restriction and extension properties for states on C$^*$-algebras
with an eye towards hyperrigidity of operator systems. We use these ideas to
provide supporting evidence for Arveson's hyperrigidity conjecture. Prompted by
various characterizations of hyperrigidity in terms of states, we examine
unperforated pairs of self-adjoint subspaces in a C$^*$-algebra. The
configuration of the subspaces forming an unperforated pair is in some sense
compatible with the order structure of the ambient C$^*$-algebra. We prove that
commuting pairs are unperforated, and obtain consequences for hyperrigidity.
Finally, by exploiting recent advances in the tensor theory of operator
systems, we show how the weak expectation property can serve as a flexible
relaxation of the notion of unperforated pairs.
