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By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$. This result is also stated and generalized by Fuchs-Hamkins-Reits in their paper Set theoretic geology.

In both of these papers, passing from $V$ to $V[G]$ some cardinals are collapsed. So my question is the following:

Question. Does any model $V$ of $ZFC$ have a cofinality preserving class generic extension $V[G]$ such that $HOD^{V[G]}=V?$

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    $\begingroup$ It seems unlikely that we will be able to answer this, in light of the fact that Foreman's conjecture is open: perhaps every forcing notion either adds a real or collapses a cardinal. In a model like that, we cannot expect to code sets into HOD by any Ord-length iteration, without collapsing cardinals or adding too many reals. $\endgroup$
    – Joel David Hamkins
    Commented Nov 3, 2015 at 12:34

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