2 problems

HOD conjecture. There is a proper class of regular uncountable cardinals $\kappa$ which are not $\omega$-strongly measurable in $\operatorname{HOD}$.

Let $V$ be a model of ${\rm ZFC}$ and let $\alpha\leq\mathop{\rm ORD}$. In another model $\overline{V}$ of ${\rm ZFC}$, write $({\mathord{\rm M}}^\alpha)^{\overline{V}}$ and…