Generalized Gaifman conjecture on non-structure over a predicate

Let $T$ be a countable complete theory, and let $P$ be a distinguished unary predicate in its vocabulary. Assume that $T$ fails the Gaifman property: there is a model $N$ of the theory of $P$ such that no model $M\models T$ satisfies $P^M=N$. Let $\lambda$ be a sufficiently large regular cardinal and let $\mu\geq\lambda$.

Generalized Gaifman conjecture. The theory $T$ has $2^\lambda$ models of cardinality $\mu$ that are pairwise non-isomorphic over $P$.

This asserts non-structure over $P$ when the Gaifman property fails. The paper separates it into stability implying the Gaifman property and instability implying non-structure; the source says that the first part is proved, while the general non-structure assertion remains a conjecture.