Representation conjecture for dense weakly selective ideals

Let $J$ be a dense $\F_{\sigma\delta}$ weakly selective ideal on $\omega$. A representation of $J$ consists of a Polish space $X$ with a countable base $\mathcal{O}$ and a $\sigma$-ideal $I$ on $X$ such that, under an identification of $\omega$ with $\mathcal{O}$, the ideal $J$ becomes $J_I$.

Representation conjecture. If $J$ is a dense $\mathbf{F}_{\sigma\delta}$ weakly selective ideal on $\omega$, then there exists a Polish space with a countable base $\mathcal{O}$ and a $\sigma$-ideal $I$ on $X$ such that under some identification of $\omega$ and $\mathcal{O}$ the ideal $J$ becomes $J_I$.

The conjecture asks for an internal characterization of ideals representable as $J_I$ for a $\sigma$-ideal on a Polish space. It is motivated by Hrušák's Category Dichotomy, which gives such a representation-related embedding result for Borel weakly selective ideals via the ideal of nowhere dense sets.