Random-chain representation of measure-one ideals

Establish whether, for every measure-one set X of reals, there exists a measure-one set Y such that every chain B contained in Y generates the same Turing ideal as some real in X.

Background

This is a stronger uniform representation question about Turing ideals generated by chains of random reals. The desired set Y would ensure that every chain selected from it is represented by a real from the prescribed full-measure set X.

References

Is it true that, for every $X\subseteq 2\omega$ with $\mu(X)=1$, there exists $Y\subseteq 2\omega$ with $\mu(Y)=1$ such that, for every chain $B\subseteq Y$, there is some $\alpha\in X$ satisfying $[B]=[\alpha]$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 1, Problem (Levin)