Topology-refinement reformulation of the strong Jayne–Rogers conjecture

Let $X$ be an analytic space, $Y$ a separable metrizable space, $1<n<m<\omega$, and $f\in\Sigma_{m,n}(X,Y)$. Topology-refinement reformulation of the strong Jayne–Rogers conjecture. The topology $\tau$ of $X$ can be refined by a topology $\tau'\subseteq\boldsymbol{\Sigma}^0_2(X,\tau)$ such that $Z=(X,\tau')$ is analytic and $f\in\Sigma_{m-1,n}(Z,Y)$. This is presented as an equivalent reformulation of the strong generalization of the Jayne–Rogers theorem; the general assertion remains open.