Shelah's definability-function cardinal conjecture

Let $\tau \subseteq \tau_1$, let $P \in \tau_1 \setminus \tau$, and let $T$ be a theory in $\tau_1$. For a model $M$, define

$$\operatorname{df}(M)=\left|\\{P:(M,P)\text{ is a reduct of a model of }T\\}\right|$$

and define

$$\operatorname{Df}(\lambda)=\sup\\{\operatorname{df}(M)^+:\\|M\\|=\lambda\\}.$$

Shelah's conjecture. If for some $\lambda\geq|\tau_1|$, $\operatorname{Df}(\lambda)>\operatorname{Ded}(\lambda)$, then for every $\lambda$, $\operatorname{Df}(\lambda)=(2^\lambda)^+$.

This concerns how failure of definability is reflected in the number of expansions of models; the supplied text gives no resolution status.