  We study the closure of the unitary orbit of a given point in the
non-commutative Choquet boundary of a unital operator space with respect to the
topology of pointwise norm convergence. This may be described more extensively
as the $\ast$-representations of the $\mathrm{C}^{\ast}$-envelope that are
approximately unitarily equivalent to one that possesses the unique extension
property. Although these $\ast$-representations do not necessarily have the
unique extension property themselves, we show that their unital completely
positive extensions display significant restrictions. When the underlying
operator space is separable, this allows us to connect our work to Arveson's
hyperrigidity conjecture. Finally, as an application, we reformulate the
classical \v{S}a\v{s}kin Theorem and Arveson's essential normality conjecture.
