Characterizability and homomorphic characterizability at singular cardinals

Let $\kappa$ be an infinite cardinal. Write $\mathcal{CH}_{\omega_1,\omega}$ for the class of cardinals characterizable by a complete Scott sentence, and $\mathcal{HCH}_{\omega_1,\omega}$ for the class of cardinals homomorphically characterizable by a complete Scott sentence. Also write $cf(\kappa)$ for the cofinality of $\kappa$.

Characterizability conjecture.

$$\kappa\in\mathcal{CH}_{\omega_1,\omega}\setminus\mathcal{HCH}_{\omega_1,\omega}\quad\Longleftrightarrow\quad cf(\kappa)=\omega.$$

The preceding results show that, under suitable cardinal-arithmetic hypotheses such as $\kappa^\omega=\kappa$, membership in $\mathcal{CH}_{\omega_1,\omega}$ implies membership in $\mathcal{HCH}_{\omega_1,\omega}$. Thus the proposed equivalence identifies countable cofinality as the only possible source of a difference between the two classes, but the statement is not established in the supplied text.