Finite-level 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 $m$-stable over $P$ for every $m<n$, but is $n$-unstable for some $n$. Let $\lambda$ be a sufficiently large regular cardinal, set $\kappa=\lambda^{+n}$, and let $\mu\geq\kappa$.

Finite-level non-structure conjecture. The theory $T$ has $2^\kappa$ models of cardinality $\mu$ that are pairwise non-isomorphic over $P$.

The source presents this as a weaker, potentially more attainable version of the higher instability conjecture. It is not proved in the supplied text.