Gandy's basis theorem says that any nonempty $\Sigma^1_1$ set $A$ contains a real $x$ with $\omega_1^x=\omega_1^{CK}$, the least nonrecursive ordinal.

Now the following question seems quite interesting to me:

Question: Is it true that for any real $z$ and nonempty $\Sigma^1_1(z)$ set $A$ containing a real $y>_hz$, $A$ must contain a real $y_0>_h z$ so that $\omega_1^{y_0}=\omega_1^z$?

Here $\geq_h$ is the hyperarithmetic reduction.