On $ω$-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 $κ\in\{ω_1, ω_2, ω_3\}$, then the restriction of the club filter on $κ\cap\mathrm{Cof}(ω)$ to HOD is an ultrafilter in HOD. This answers Question 4.11 of [BNH23] raised by Ben-Neria and Hayut.
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