Shelah's eventual model-counting bound for abstract elementary classes

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class (AEC), and let

$$\delta=(2^{\operatorname{LS}(\operatorname{\mathcal{K}})})^+.$$

Assume that $\operatorname{\mathcal{K}}$ has at least one model but fewer than the maximal number of models in some cardinal $\lambda>\beth_\delta$. Shelah's conjecture. The number of nonisomorphic models of size $\aleph_\alpha$ is bounded by

$$\beth_\delta(\aleph_0+|\alpha|)$$

for each ordinal $\alpha$. This is a proposed model-counting bound in the theory of abstract elementary classes, motivated by the Main Gap programme: a structural decomposition should constrain the number of models, while failure of structure can yield the maximal number. The source presents the statement as a conjecture from the late 1990s; no resolution is given here.