Weak representability conjecture for measurable preimage functions

Let $n$ and $m$ be the parameters for the corresponding Borel measurability levels, and let $\boldsymbol{X}$ and $\boldsymbol{Y}$ be Polish represented spaces. Write $\mathfrak{b}$ for the computable endofunctor capturing Borel measurability, and let $\mathcal{C}_{-1}$ denote the space of continuously represented preimage functions. Weak representability conjecture. The underlying sets of

$$\mathcal{C}_{-1}(\mathcal{O}^{(n)}(\mathbf{Y}), \mathfrak{b}\left (\mathcal{O}^{(m)}(\mathbf{X})\right ))$$

and

$$\mathcal{C}_{-1}(\mathcal{O}^{(n)}(\mathbf{Y}), \left (\mathcal{O}^{(m)}(\mathbf{X})\right ))$$

contain the same elements. This conjecture asks whether adding the Borel-measurability endofunctor changes the underlying collection of such preimage functions, beyond changing its represented-space structure.