Graph complexity conjecture for finite-level decomposable functions

Let $X$ and $Y$ be separable metrizable spaces with $X$ analytic, and let $\widetilde{Y}$ denote the completion used for the graph formulation. For $1<n<\omega$, let $\mathsf{D}_n(X,Y)$ be the class of $n$-decomposable functions considered in the paper. Graph complexity conjecture. Every function in $\mathsf{D}_n(X,Y)$ has graph in $\boldsymbol{\Sigma}^0_n(X\times\widetilde{Y})$. The paper states that this is equivalent to the weak generalization of the Jayne–Rogers theorem, so its resolution is tied to the finite-level decomposition problem.