The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal.

Question.Assume $\kappa$ is a strongly compact cardinal. Is there an inner model $M$ of the universe such that $M \models$``$\kappa$ is strongly compact and the set of measurable cardinals below $\kappa$ is unbounded in $\kappa$'' $?$