  Let $A$ be a unital $C^*$-algebra generated by some separable operator system
$S$. More than a decade ago, Arveson conjectured that $S$ is hyperrigid in $A$
if all irreducible representations of $A$ are boundary representations for $S$.
Recently, a counterexample to the conjecture was found by Bilich and Dor-On. To
circumvent the difficulties hidden in this counterexample, we exploit some of
Pedersen's seminal ideas on noncommutative measurable structures and establish
an amended version of Arveson's conjecture. More precisely, we show that all
irreducible representations of $A$ are boundary representations for $S$
precisely when all representations of $A$ admit a unique "tight" completely
positive extension from $S$. In addition, we prove an equivalence between
uniqueness of such tight extensions and rigidity of completely positive
approximations for representations of nuclear $C^*$-algebras, thereby extending
the classical principle of Korovkin--Saskin for commutative algebras.
