Example with right-open amalgamation spectrum

Question

Let $\phi$ be an $L_{\omega_1,\omega}$ sentence. The amalgamation spectrum of $\phi$ is the set of all cardinals $\kappa$ such that the models of $\phi$ of size $\kappa$ satisfy amalgamation.

Question: Is there a known example where the amalgamation spectrum is right-open? E.g. of the form $[\kappa,\lambda)$, where $\lambda$ is a limit cardinal.

Answer 1

In this paper we were able to prove that there exists some $\psi\in L_{\omega_1,\omega}$ and it is consistent that the amalgamation spectrum of $\psi$ is consistently equal to $[\aleph_1,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. This (consistently) answers the question.