Main gap and super-limit spectrum conjecture for abstract elementary classes

Let ${\mathfrak k}$ be an abstract elementary class. Main gap and super-limit conjecture. (1) The main gap theorem holds for $K_{\mathfrak k}$ for all sufficiently large $\lambda$. (2) The class

$$\operatorname{\sup\!\!\text{-}\lim}_{\mathfrak k}=\{\lambda: \text{there is a super-limit }M\in K^{\mathfrak k}_\lambda\}$$

is nice; for example, it contains every sufficiently large cardinal or contains no sufficiently large cardinal. The source labels this a much harder conjecture and gives no resolution.