According to this paper, by Vaananen, the $LS$ number for $2^{nd}$ order logic is given by "the supremum of $Π_{2}$-definable ordinals", where "The Lowenheim-Skolem number $LS(L)$ of $L$ is the smallest cardinal $\kappa$ such that if a theory $T\subseteq L$ has a model, it has a model of cardinality $< \max(\kappa, |T|)$."

So I'm wondering if the $LS$ number is larger for higher-order logics (compared to $2^{nd}$ order logic)?