Collapsing the cardinals of $HOD$

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

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