Woodin's HOD conjecture

Let $\kappa$ be an uncountable regular cardinal, and let $S$ be a stationary subset of $\kappa$. We say that $\kappa$ is strongly measurable in $\operatorname{HOD}$ with respect to $S$ if there is some $\eta<\kappa$ such that $(2^\eta)^{\operatorname{HOD}}<\kappa$ and no partition of $S$ into $\eta$ many sets, all stationary in $V$, belongs to $\operatorname{HOD}$. We say that $\kappa$ is $\omega$-strongly measurable in $\operatorname{HOD}$ when this holds for $S=\kappa\cap\operatorname{Cof}(\omega)$.

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

This is a combinatorial assertion about $\operatorname{HOD}$ and $V$, rather than an assertion requiring inner-model constructions. The supplied text attributes the conjecture to Woodin and describes significant consequences if it holds, but gives no resolution.