On $\omega$-strongly measurable cardinals in ${\mathbb{P}}_{\max }$ extensions

We show that in the ${\mathbb{P}}_{\max }$ extension of a certain Chang-type model of determinacy, if $\kappa \in \{{\omega }_1,{\omega }_2,{\omega }_3\}$, then the restriction of the club filter on $\kappa \cap \mathrm{C}\mathrm{o}\mathrm{f}(\omega )$ to HOD is an ultrafilter in HOD. This answers Question 4.11 of [BNH23] raised by Ben-Neria and Hayut.