Gaifman's conjecture on categoricity over a predicate and nulldimensionality

Let $T$ be a countable complete first-order theory with a distinguished predicate $P$. Say that $T$ is categorical over $P$ when it has at most one model, up to isomorphism over $P$, with any prescribed $P$-part. Let $T$ be nulldimensional over $P$ in the sense used in the source, and let a u.l.a. $\mathcal P^-(n)$-system mean the corresponding system defined in the paper. Gaifman's conjecture. The class of $\aleph_1$-saturated models of $T$ is absolutely categorical over $P$ if and only if $T$ is nulldimensional, and every u.l.a. $\mathcal P^-(n)$-system is stable over $P$. The conjecture is the proposed completion of the equivalence results summarized immediately beforehand: the source explains that the relevant non-structure implication is fully proved only for $n\leq 3$, while the general case is believed to follow from Shelah's argument but is not fully established there.