Large-cardinal dichotomy for abstract elementary classes

Let ${\mathfrak k}$ be an abstract elementary class. Large-cardinal dichotomy conjecture. For a closed unbounded class $C$ of cardinals, the alternatives stated in the source should hold: (1) the promised conclusion of Theorem 28 can be obtained with $C$ an end segment of $\{\mu:\mu=\beth_{1,\mu}\}$, or with $C=\{\mu:\mu=\beth_{2,\mu}\}$; and (2) for every such ${\mathfrak k}$, either

$$M\in K^{\mathfrak k}_\lambda\wedge\lambda\in C\wedge {\rm cf}(\lambda)=\aleph_0\Rightarrow \Upsilon^{\rm or}_\lambda[\mathfrak k_M]\neq\emptyset,$$

or

$$\lambda\in C\wedge {\rm cf}(\lambda)=\aleph_0\Rightarrow \dot I(\lambda,K_{\mathfrak k})\geq 2^\lambda\text{ (or at least }\lambda^+\text{)}.$$

The source says that part (1) is intended for a continuation; the claims are therefore unresolved.