Stationarity dichotomy conjecture for abstract elementary classes

Let ${\mathfrak k}$ be an abstract elementary class and define

$$C^{\rm fp}_{\aleph_0}=\{\lambda:\lambda=\beth_\lambda\text{ and }{\rm cf}(\lambda)=\aleph_0\}.$$

Set

$$S_1=\{\lambda\in C^{\rm fp}_{\aleph_0}:\dot I(\lambda,\mathfrak k)<\lambda\},\qquad S_2=\{\lambda\in C^{\rm fp}_{\aleph_0}:\dot I(\lambda,\mathfrak k)\geq\lambda\}.$$

Stationarity dichotomy conjecture. The classes $S_1$ and $S_2$ are not both stationary. A weaker version replaces $S_2$ by the class of $\lambda\in C^{\rm fp}_{\aleph_0}$ such that every $M\in K^{\mathfrak k}_\lambda$ has $\leq_{\mathfrak k}$-extensions of every cardinality greater than $\lambda$. The source presents these as conjectures motivated by controlling the categoricity spectrum, with no resolution stated.