Grossberg's eventual amalgamation conjecture for abstract elementary classes

Let $\lambda$ be a cardinal. For an abstract elementary class (AEC) $\boldsymbol{K}$, write $\operatorname{LS}(\boldsymbol{K})$ for its Löwenheim–Skolem number, and say that $\boldsymbol{K}$ has the $\mu$-amalgamation property when amalgamation holds in cardinality $\mu$. Grossberg's conjecture. For every $\lambda$, there is a cardinal $\mu(\lambda)$ such that every AEC $\boldsymbol{K}$ with the $\mu(\operatorname{LS}(\boldsymbol{K}))$-amalgamation property has the $\lambda$-amalgamation property for all $\lambda\geq\mu(\operatorname{LS}(\boldsymbol{K}))$. The paper presents this as a question about eventual amalgamation in AECs; the supplied excerpt does not state whether the conjecture is resolved.