The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$.

Although this topology is definitely less nice than the standard one - for example, it is non-metrizable - it satisfies the *strong Choquet property*: this is the statement that player I has a winning strategy in a certain topological game. From the strong Choquet property we can deduce many results whose proof would follow easily from metrizability, so in some sense the Gandy-Harrington topology is "close enough" to metrizable. One particularly nice result we can show is: if $A$ is a nonempty lightface $\Sigma^1_1$ subset of $\omega^\omega$, then $A$ has an element $x$ such that $\mathcal{O}^x\equiv_T\mathcal{O}\oplus x$.

A natural question now is to ask about topologies generated by lightface sets further up the projective hierarchy; e.g., let $\tau_k$ be the topology on $\omega^\omega$ generated by lightface $\Sigma^1_k$ sets. And similarly, we can define $\tau_\Gamma$ for any pointclass $\Gamma$ (although we're probably only interested in, say, Spector pointclasses). Unfortunately, for $k>1$ the strong Choquet property fails badly in $\tau_k$!

My question is:

Is there a weakening of the strong Choquet property that holds for $\tau_2$? (Or more generally for $\tau_\Gamma$ for nice enough pointclass $\Gamma$.)

Relatedly, for a real $x$ let $\mathcal{O}_2^x$ be the set of $x$-computable codes for nonempty $\Sigma^1_2$ sets, and let $\mathcal{O}_2=\mathcal{O}_2^\emptyset$.

Is it the case that every lightface $\Sigma^1_2$ subset of $\omega^\omega$ contains an $x$ with $\mathcal{O}_2^x\equiv_T\mathcal{O}_2\oplus x$?

*I've tagged "set-theory" and "inner-model-theory" because it seems reasonable to me that answers might depend on structural hypotheses about the universe; feel free to remove either tag if this seems bonkers.*