Categoricity transfer conjecture for abstract elementary classes

Let ${\mathfrak k}=(K_{\mathfrak k},\leq_{\mathfrak k})$ be an abstract elementary class, with Löwenheim–Skolem–Tarski number ${\rm LST}_{\mathfrak k}$. Categoricity transfer conjecture. If $K_{\mathfrak k}$ is categorical in some cardinal $\lambda$ sufficiently large compared with ${\rm LST}_{\mathfrak k}$, then $K_{\mathfrak k}$ is categorical in every $\mu\geq\lambda$. This is a central categoricity-transfer problem for abstract elementary classes; the source notes partial progress via good frames but does not claim a general proof.