# A question related to Woodin's $HOD$ conjecture

Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists $\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that there is no partition of $S = \{\alpha < \kappa: \text{cf}(\alpha) = \omega\}$ into $\lambda$ many sets $\langle S_{\alpha}: \alpha < \lambda\rangle \in \text{HOD}$ such each set $S_{\alpha}$ is stationary in $V$.

It is not known if the successor of a singular strong limit of uncountable cofinality can be $\omega$-strongly measurable in $\text{HOD}.$

Now my question is about countable cofinality:

Question 1. Is it consistent that the successor of some singular strong limit cardinal of countable cofinality is $\omega$-strongly measurable in $\text{HOD}$?

If consistent, would you please give some references for it.

I asked the question from Prof. Woodin. The answer is yes, assuming the consistency of Axiom I0. It has appeared as Lemma 190 on page 298 of Woodin's paper Suitable Extender Models I''.