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.