Superlimit-or-maximal-nonstructure conjecture for abstract elementary classes

Let ${\mathfrak k}$ be an abstract elementary class and put $\kappa={\rm LST}_{\mathfrak k}+|\tau_{\mathfrak k}|$. Superlimit-or-nonstructure conjecture. At least one of the following holds: (a) whenever $\lambda=\beth_{1,\lambda}>\kappa$ has cofinality $\aleph_0$, there is a nonempty family $\Upsilon^{\rm sor}_\kappa[M,\mathfrak k]$; or (b) whenever $\lambda=\beth_{1,\lambda}>\kappa$ has cofinality $\aleph_0$, $\dot I(\lambda,K_{\mathfrak k})=2^\lambda$. The source gives this as a concluding conjectural dichotomy, with no resolution stated.