Strong representability conjecture for measurable preimage functions

Let $n$ and $m$ be the parameters for the corresponding Borel measurability levels, and let $\mathbf{X}$ and $\mathbf{Y}$ 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. Strong representability conjecture. The spaces

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

and

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

are computably isomorphic. This strengthens the weak conjecture from equality of underlying sets to an isomorphism of the represented spaces, and its resolution would clarify how Borel measurability interacts with computable representations of preimage functions.