Say that a set is ordinal$_{\kappa\lambda}$-definable if it is definable by a formula in the infinitary language $\mathcal{L}_{\kappa\lambda}$ with parameters from $\mathsf{On}$. Let $HOD_{\kappa\lambda}$ be the class of hereditarily ordinal$_{\kappa\lambda}$-definable sets. Definitionally, $HOD_{\omega\omega} = HOD$; by a simple argument given in the citation below, $HOD_{\infty\infty}=V$.

What can we say about $HOD_{\kappa\lambda}$ for other values of $\kappa$ and $\lambda$? Some questions that obviously arise: do we have $\mathsf{Con}(HOD_{\kappa^+\omega}\neq HOD_{\kappa\omega})$ for all $\kappa$? Do we have $\mathsf{Con}(HOD_{\kappa^+\lambda}\neq HOD_{\kappa\lambda})$ for all $\kappa,\lambda$? Do we have $\mathsf{Con}(HOD_{\kappa\lambda^+}\neq HOD_{\kappa\lambda})$ for all $\kappa,\lambda$?

(This question is inspired by the discussion at Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem).)