Combining chains of random reals

Determine whether every chain of random reals in the Turing degrees can be combined into a single random real, and whether every Turing ideal generated by random reals can be generated by the columns of a single random real.

Background

The problem concerns the structure of Turing ideals generated by sufficiently random reals. It asks whether countably many random degrees can be represented by one random real, either directly through a combined chain or through its columns.

References

Are the following true? (1) Can every chain of random reals in the Turing degrees be combined into a single random real? (2) Can every Turing ideal generated by random reals be generated by the columns of a single random real?

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

Is there a chain $B\subseteq\mathrm{ML}(\emptyset')$ such that $[B]\neq[\alpha]$ for every $\alpha\in\mathrm{ML}(\emptyset')$?

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