Collapsing the cardinals of $HOD$

Assuming that $GCH$ holds and $\kappa$ is $\kappa^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $\kappa$ remains strongly inaccessible and $(\alpha^+)^{HOD} < \alpha^+$ for every infinite cardinal $\alpha < \kappa$. In particular the rank-initial segment $W_\kappa$ is a model of ZFC in which $(\alpha^+)^{HOD} < \alpha^+$ for every infinite cardinal $\alpha$.