On $({\mathrm{\Sigma }}_1^2{)}^{uB}$ Absoluteness Between V and HOD

We put together Woodin’s ${\mathrm{\Sigma }}_1^2$ basis theorem of AD${}^{+}$ and Vopěnka’s theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $({\mathrm{\Sigma }}_1^2{)}^{uB}$ statement that is true in $V$ is true in $HOD$. Moreover, this is true even if we allow a parameter $C\subseteq \mathbb{R}$ such that $C$ and its complement have scales that are $OD$ and universally Baire. We also investigate whether $({\mathrm{\Sigma }}_1^2{)}^{uB}$ statements are upwards absolute from $HOD$ to $V$ under large cardinal hypotheses, observing that this is true if $HOD$ has a proper class of Woodin cardinals. Finally, we discuss $({\mathrm{\forall }}^{\mathbb{R}})({\mathrm{\Sigma }}_1^2{)}^{uB}$ absoluteness and conclude that this much absoluteness between $HOD$ and $V$ cannot be implied by any large cardinal axiom consistent with the axiom ``$V=$ Ultimate $L$’'.