Shelah proved that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is $X \subseteq \kappa$ such that $P \kappa \subseteq \mathrm{HOD}_X$. We prove that consistently this fails badly for strong limit $\kappa$ of countable cofinality. (Joint work with Sy Friedman, Menachem Magidor, Assaf Ronot and Dima Sinapova).