Finite-level decomposability conjecture for Borel functions

Let $m,n\in\omega$. For a class of Borel functions, $\mathbf{dec}_{n+1}\mathcal{F}$ denotes functions decomposable into countably many $\mathcal{F}$-functions with $\mathbf{\Delta}^0_{n+1}$ domains. For separable metric spaces with analytic domain, let $\mathbf{\Sigma}_{m+1,n+1}$ denote the corresponding class of Borel functions and let $\mathbf{\Sigma}_{1,n-m+1}$ denote the class of $\mathbf{\Sigma}^0_{n-m+1}$-measurable functions. Finite-level decomposability conjecture. On separable metric spaces with analytic domain, $\mathbf{\Sigma}_{m+1,n+1}=\mathbf{dec}_{n+1}\mathbf{\Sigma}_{1,n-m+1}$ holds; equivalently, the $\mathbf{\Sigma}_{m+1,n+1}$ functions are precisely the $\mathbf{\Delta}^0_{n+1}$-piecewise $\mathbf{\Sigma}^0_{n-m+1}$-measurable functions at all finite levels $m,n\in\omega$. This proposed identity generalizes the Jayne–Rogers and Semmes decompositions at finite Borel levels. The surrounding discussion cites prior work establishing low-level cases and dichotomies, while the all-finite-level assertion is presented as an expectation rather than as a theorem.