  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
$\alpha$.
