Definability of Borel-base ideals on positive restrictions

Determine whether every $\sigma$-ideal with a Borel base has a restriction to some positive set that is $\mathbf{\Delta}^1_2$ on $\mathbf\Sigma^1_1$, including when the ideal has the universality property, induces a proper forcing, or arises from an $F_\sigma$ relation.

Background

The paper constructs an FσF_\sigma relation with the total universality property whose induced ideal is not Δ21\mathbf{\Delta}^1_2 on analytic sets. It also constructs an example that is definable at this level but is not a game ideal. These examples motivate the question of whether definability might nevertheless hold after restricting an ideal to a suitable positive set, especially under additional structural assumptions.

References

For a $\sigma$-ideal with a Borel base, is its restriction to some positive set $\mathbf\Delta1_2$ on $\mathbf\Sigma1_1$? What if we assume the universality property or that it results in a proper forcing? What about $F_\sigma$ relations?

— Universal domination and idealized forcing  (2608.18964 - Schilhan, 19 Aug 2026) in Question labelled \ref{quest:univdef}, Section 7, immediately after Theorem \ref{thm:univnotimplydef}

It seems to be a difficult problem to characterise $\omega$-refining Borel sets and we do not know if the ideal is $\mathbf\Delta1_2$ on $\mathbf\Sigma1_1$.

— Universal domination and idealized forcing  (2608.18964 - Schilhan, 19 Aug 2026) in Paragraph beginning “There are a few more things of interest...” in Section 7, subsection “Refining forcing”