No represented space characterizes limit computability by closed choice

Let $(X,\delta)$ be a represented space, let ${\rm C}_X$ denote closed choice on $X$, and let $\lim$ denote the limit operation on Baire-space names. Non-characterization conjecture. There is no represented space $(X,\delta)$ such that

$${\rm C}_X\mathop{\equiv_{\mathrm{W}}}\lim.$$

Closed choice on natural spaces characterizes several classes of computable functions, but limit computable functions are not closed under composition, unlike the classes discussed in the surrounding results. The conjecture asserts that no represented space yields limit computability exactly through Weihrauch-equivalent closed choice.