Approximate categoricity of beautiful pairs of randomized theories

Let $Q$ be an $\aleph_0$-categorical, $\aleph_0$-stable theory in a countable language, and let $Q_P$ denote the theory of beautiful pairs of models of $Q$. Approximate categoricity conjecture. The theory $Q_P$ is approximately $\aleph_0$-categorical. This conjecture proposes an approximate analogue of the characterization of strongly finitely based theories by $\aleph_0$-categoricity of their theories of beautiful pairs; it is motivated by examples where exact $\aleph_0$-categoricity fails but approximate $\aleph_0$-categoricity holds. Its general status is not resolved in the supplied source.