A question related to Woodin's $HOD$ conjecture

Question

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.

Answer 1

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''.