In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement:

"For second order logic, $LS(L^{2})$ [the Löwenheim-Skolem number for second order logic--my comment] is the supremum of $\Pi_{2}$-definable ordinals..., which means that it exceeds the first measurable, the first $\kappa^{+}$-supercompact $\kappa$, and the first huge cardinal if they exist ["the Löwenheim-Skolem number $LS(L^{2})$ of second order logic $L^{2}$ is the smallest cardinal $\kappa$ such that if a theory $T$$\subset$$L^{2}$ has a model, it has a model of cardinality$\le$max($\kappa$,$|T|$)", and "$L^{2}$ extends first order logic with quantifiers of the form $\exists$$R$$\phi$($R$,$x_0$,...,$x_{{n}-1}$), where the second order variable $R$ ranges over n-ary relations on the universe for some fixed n"--my comment also but substantially quoting the authors]".

Assume that the first measurable, the first $\kappa^{+}$-supercompact $\kappa$, and the first huge cardinals exist. What type of large cardinal, then, is $LS(L^{2})$? If the answer is known, please provide the reference.

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