The answer to the question is yes. In fact, it is possible to prove something stronger:

Theorem (Omer Ben-Neria, Spencer Unger) Assuming the existence of suitable large cardinals, there exists a model $V$ of $ZFC$ which contains a club of cardinals all of them are regular in $HOD$ of $V$.

Their work is under preparation.

Remark. Their result is optimal in the sense that we can not hope to build a model in which all infinite cardinals are regular in $HOD$, as for example $\aleph_\omega$ is always singular in $HOD$.

The answer to the question is yes. In fact, it is possible to prove something stronger:

Theorem (Omer Ben-Neria, Spencer Unger) Assuming the existence of suitable large cardinals, there exists a model $V$ of $ZFC$ which contains a club of cardinals all of them are regular in $HOD$ of $V$.

Their work is under preparation.

Remark. Their result is optimal in the sense that we can not hope to build a model in which all infinite cardinals are regular in $HOD$, as for example $\aleph_\omega$ is always singular in $HOD$.

Update: The paper by Omer Ben-Neria and Spencer Unger is now available:

Homogeneous changes in cofinalities with applications to HOD