Shelah's conjecture on models at the first uncountable beth cardinal

Let ${\aleph_{\omega_1}}$ be the cardinal at index $\omega_1$, and let ${\mathcal{L}_{\omega_1,\omega}}$ be the infinitary logic allowing countable conjunctions and finite quantifier strings. Assume

$$\aleph_{\omega_1}<2^{\aleph_0}.$$

Shelah's conjecture. Every ${\mathcal{L}_{\omega_1,\omega}}$-sentence with a model of cardinality $\aleph_{\omega_1}$ has a model of cardinality $2^{\aleph_0}$.

This is presented as a conjecture attributed to Shelah. The surrounding discussion says that the answer is unknown because it is not known whether the relevant closure set captures all characterizable cardinals.