Shelah's generalized categoricity conjecture for $L_{\lambda^+,\omega}$

Let $\psi$ be a sentence of $L_{\lambda^+,\omega}$ in a language of size $\lambda$. Categoricity means having exactly one model, up to isomorphism, in a given cardinality. Shelah's generalized categoricity conjecture. If $\psi$ is categorical in some $\mu\geq\beth_{(2^\lambda)^+}$, then $\psi$ is categorical in all $\mu\geq\beth_{(2^\lambda)^+}$. The statement is presented as a generalization of Shelah's earlier conjecture; the supplied text does not state whether it has been resolved.