*I'd like to close a gap left open in an old question of mine; I've tweaked the terminology to be a bit nicer.*

For a (boldface) pointclass $\Gamma$ and a payoff set $G\subseteq\omega^\omega$, say that $G$ is $\Gamma$-**narrow** iff there is some equivalence relation $E\in\Gamma$ such that $E$ has $\le\omega_1$-many classes and $G$ is $E$-invariant. Gabe Goldberg showed that, provably in $\mathsf{ZFC+\neg CH}$, there is a $(\Sigma^1_1\wedge\Pi^1_1)\vee(\Sigma^1_1\wedge\Pi^1_1)$-narrow set whose associated Gale-Stewart game is undetermined.

My question is whether this bound can be improved. For example, is there $\mathsf{ZFC+\neg CH}$-provably a $\Pi^1_1\vee\Sigma^1_1$-narrow undetermined game? What about $\Sigma^1_1\wedge\Pi^1_1$?

Briefly, the motivation for this is that when $\mathsf{CH}$ fails, an assumption of $\Gamma$-narrowness prevents an easy construction of an undetermined game via diagonalization against strategies since there are more strategies than there are basic facts needed to determine the whole payoff set. Producing $\mathsf{ZFC+\neg CH}$ examples of $\Gamma$-narrow undetermined games, especially for "tame" $\Gamma$s, seems to require meaningfully more effort than straight diagonalization. For this reason I'd also be interested in the situation of "$\le\omega_1$" is replaced with "$<2^{\aleph_0}$," although I am primarily interested in the precise version above.