The cofinality characterization of non-homogeneously characterizable cardinals

Let $\kappa$ be a characterizable cardinal. A cardinal is homogeneously characterizable if it is characterized by a homogeneous Scott sentence, and $\operatorname{cf}(\kappa)$ denotes its cofinality.

Cofinality conjecture. For $\kappa$ a characterizable cardinal, $\kappa$ is not homogeneously characterizable if and only if

$$\operatorname{cf}(\kappa)=\omega.$$

This would characterize exactly which characterizable cardinals fail to be homogeneously characterizable. The surrounding discussion presents it as an open question, noting that it is consistent that all cardinals in ${\mathcal{CH}_{\omega_1,\omega}}\setminus{\mathcal{HCH}_{\omega_1,\omega}}$ have cofinality $\omega$, but does not establish the stated equivalence.