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.



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


