Unique choice is Weihrauch-equivalent to choice on Baire space

Let ${\rm UC}_X$ denote the unique closed choice operation on a represented Hausdorff space $X$, and let ${\rm C}_X$ denote closed choice on $X$. For Baire space { obreak\begin{\mathbb{N}}^{ obreak\begin{\mathbb{N}}}}, the unique-choice conjecture.

$${\rm UC}_{{\mathbb{N}}^{\mathbb{N}}}\mathop{\equiv_{\mathrm{W}}}{\rm C}_{{\mathbb{N}}^{\mathbb{N}}}.$$

Unique choice is always Weihrauch-reducible to full choice, and the corresponding equivalence is proved for ${\mathbb{N}}$. The conjecture asks whether the same phenomenon holds for Baire space.