Higher instability non-structure conjecture over a predicate

Let $T$ be a countable theory and let $P$ be a distinguished unary predicate. Assume that $T$ is $n$-unstable over $P$ for some $n$. For every sufficiently large regular cardinal $\lambda$ and every $\mu\geq\lambda$, consider models of $T$ of cardinality $\mu$, with isomorphism taken over $P$.

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

This is the paper's restatement of the non-structure direction of the generalized Gaifman conjecture. The source presents it as unproved and notes that proving it may be challenging.