"Clubbiness" of $\Pi^1_n$ sets

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I'm sure this is just my google-fu failing me, but: what are sufficient large cardinal axioms to guarantee "Every (boldface) $\Pi^1_n$ set of countable ordinals contains or is disjoint from a club subset of $\omega_1$"?

EDIT: To clarify, I'm asking about the consistency strength over ZFC.

Here's a very silly upper bound: suppose $L(\mathbb{R})$ is a model of AD; note that every $\Pi^1_n$-sentence with real parameters is absolute between $L(\mathbb{R})$ and $V$. Then let $A\in V$ be a $\Pi^1_n$-set of countable ordinals, via the formula (with real parameters) $\varphi$. By the absoluteness assumption, $\varphi^{L(\mathbb{R})}=A$, so $A\in L(\mathbb{R})$. And since $L(\mathbb{R})\models AD$, $L(\mathbb{R})$ thinks $A$ contains or is disjoint from a club. But inner models compute club-ness correctly, so we're done.

This seems massively overkill to me, though - what is the right bound?