Approximate categoricity for randomized theories with disintegrated geometry

Let $\mathcal{T}$ be the class of $\aleph_0$-categorical, $\aleph_0$-stable classical theories in countable languages, and let $\mathcal{D}\subseteq\mathcal{T}$ consist of those theories $T$ such that every strictly minimal set interpretable in $T$ has disintegrated geometry. Write $T^R$ for the randomized theory of $T$, and $(T^R)_P$ for the theory of beautiful pairs of models of $T^R$. Approximate categoricity conjecture for $\mathcal{D}$. If $T\in\mathcal{D}$, then $(T^R)_P$ is approximately $\aleph_0$-categorical. The source establishes the opposite behavior for $T\in\mathcal{T}\setminus\mathcal{D}$ and proves the claim for the theory of infinite sets, together with stability results under several permutation-group constructions. The exact extent of the class for which approximate $\aleph_0$-categoricity holds remains open.