In the paper "A reflection phenomenon in descriptive set theory" by J.P. Burgess, it is stated:

Moschovakis [11] has proved the following important Uniform Boundedness Theorem: There is a continuous function $\Gamma: Y \rightarrow Z$ such that for all $y \in Y$, if $U_y \subseteq W$, then $\Gamma(y) \in W$ and $z \prec \Gamma(y)$ for all $z \in U_y$,

where $Y=2^{\omega \times \omega}$, $Z$ is the set of codes for linear orders on $\omega$ and $W=WO$.

But the reference [11] hadn't yet appeared and I'm able to find only the boundedness theorem in [11], but not the uniform version. How can we show this theorem? Have you ever seen this uniform version somewhere?