Grossberg's two-cardinal categoricity conjecture for AECs

Let ${\bf K}$ be an abstract elementary class (AEC), and let $\lambda\geq\operatorname{LS}({\bf K})$, where $\operatorname{LS}({\bf K})$ is its Löwenheim–Skolem number. Write $I(\mu,{\bf K})$ for the number of models in ${\bf K}$ of cardinality $\mu$, up to isomorphism.

Grossberg's conjecture. If

$$I(\lambda,{\bf K})=I(\lambda^+,{\bf K})=1,$$

then

$$K_{\lambda^{++}}\neq\emptyset.$$

The conjecture asserts that categoricity in two successive cardinals implies the existence of a model in the next cardinal. It was suggested by Grossberg in 1994, motivated by earlier work, and is presented here as an application concerning the upward transfer of categoricity for abstract elementary classes.