The theorem 1.1 of the following paper:
Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic.
states that:
Theorem: Assume $V\models ZFC+GCH+~\text{There exists a}~(\kappa+4)-\text{strong cardinal}~\kappa$ then there is a generic extension $W$ of $V$ such that:
(1) $\kappa$ remains inaccessible in $W$.
(2) $V_{\kappa}^W=W_\kappa\models ZFC+\forall \lambda, ~2^{\lambda}=\lambda^{+3}$.
(3) $HOD^{W_{\kappa}}\models GCH$.
The theorem shows the consistency of having a model of $ZFC$ (like $W_{\kappa}$) in which $GCH$ fails everywhere (in a finite gap form) but in its $HOD$, $GCH$ holds.
In a remark after the theorem the author says:
Remark: In fact it suffices to have a Mitchell increasing $\kappa^+$-sequence of extenders, each of which is $(\kappa + 3)$-strong. Thus for the conclusion of Theorem 1.1 it suffices to have a cardinal $κ$ such that $o(κ) = \kappa^{+3}+\kappa^+$.
My question is:
Question: Is $o(κ) = \kappa^{+3}+\kappa^+$ necessary for the proof of theorem 1.1. or one could hopefully reduce it to something weaker?
I guess here some sort of core model argument is needed to prove such a result but maybe there are other methods to ensure that the assumption could not be weaker.
