Debs–Saint Raymond rank characterization conjecture for analytic ideals

Let $\mathcal{I}$ be an analytic ideal. For $0<\alpha<\omega_1$, let $\operatorname{rk}(\mathcal{I})$ denote the least $\beta<\omega_1$ such that some $\boldsymbol{\Sigma}^0_{1+\beta}$ set separates $\mathcal{I}$ from its filter dual, and let $\mathrm{Fin}_\alpha$ be the ideal of rank $\alpha$. Say that an ideal contains an isomorphic copy of $\mathrm{Fin}_\alpha$ if $\mathrm{Fin}_\alpha\sqsubseteq\mathcal{I}$ in the sense defined above.

Debs–Saint Raymond conjecture. For every analytic ideal $\mathcal{I}$ and $0<\alpha<\omega_1$,

$$\operatorname{rk}(\mathcal{I})\geq\alpha\quad\Longleftrightarrow\quad \mathrm{Fin}_\alpha\sqsubseteq\mathcal{I}.$$

This conjecture proposes a combinatorial characterization of the Borel separation rank of analytic ideals: having rank at least $\alpha$ should be equivalent to containing the canonical ideal $\mathrm{Fin}_\alpha$ up to an isomorphic copy. The source presents it as a conjecture of Debs and Saint Raymond; no resolution is supplied here.