Reverse mathematics on lightface $\Pi^1_1$-uniformization for unary relation

Question

It is known that the following form of $\Pi^1_1$-uniformization is equivalent to $\Pi^1_1$-Comprehension over $\mathsf{ATR}_0$ (cf. VI.2.6 of Simpson's book) :

(Kondo's uniformization theorem) For each $\Pi^1_1$-formula $\psi(X,Y)$, we can find another $\Pi^1_1$ formula $\hat{\psi}(X,Y)$ such that the following holds:

1. $\forall X,Y [\hat{\psi}(X,Y)\to\psi(X,Y)].$
2. $\forall X, Y, Z [\hat{\psi}(X,Y)\land \hat{\psi}(X,Z)\to Y=Z]$.
3. $\forall X, Y [\psi(X,Y) \to \exists Z \hat{\psi}(X,Z)]$.

Question. Is the following form of $\Pi^1_1$-uniformization equivalent to $\Pi^1_1$-Comprehension over $\mathsf{ATR}_0$, or provable from a weaker subsystem like parameter-free $\Pi^1_1$-comprehension or $\mathsf{ATR_0}$ plus $\Pi^1_1$-Transfinite Induction?

(Uniformization for a unary lightface $\Pi^1_1$ formula) For each $\Pi^1_1$-formula $\psi(X)$ with all free variables displayed, we can find another $\Pi^1_1$ formula $\hat{\psi}(X)$ (also, with all free variables displayed) such that the following holds:

1. $\forall X [\hat{\psi}(X)\to\psi(X)].$
2. $\forall X, Y [\hat{\psi}(X)\land \hat{\psi}(Y)\to X=Y]$.
3. $\forall X [\psi(X) \to \exists Y \hat{\psi}(Y)]$.

The usual proof for the $\Pi^1_1$-uniformization theorem (one using scales for $\Pi^1_1$ sets, also provided in Simpson's book) proves Kondo's uniformization theorem, so the usual proof always requires $\Pi^1_1$-Comprehension. Simpson's proof of $\Pi^1_1$-Comprehension from Kondo's uniformization theorem seems to heavily use the uniformization for binary relations, so I do not see if the $\Pi^1_1$-uniformization for unary formula implies the existence of, for example, the hyperjump of $0$.

Answer 1

I will give this another attempt. For each $\Pi^1_1$ formula $\phi_i$ with one set variable let $\widehat{\phi_i}$ be a $\Pi^1_1$ formula which uniformizes $\phi_i$ and $X_i$ be the unique set satisfying $\widehat{\phi_i}$. Let $Z=\oplus_{i\in\omega}X_i$. Let $\mathcal{S}\subseteq\mathcal{P}(\omega)$ be a $\beta $-model which contains $Z$ but does not contain $\mathcal{O}(Z)$. Such a model can be obtained relativizing Theorem VIII.6.8 in Simpson's. So $\mathcal{S}$ is a model of $\textbf{ATR}_0$ but not a model of $\Pi^1_1\textbf{-CA}_0$. Any $\Pi^1_1$ formula with one set variable will be some $\phi_i$. By construction we have that $\widehat{\phi_i}$ will uniformize $\phi_i$ in $\mathcal{S}$.