2-random ideal with no relative randomness

Construct a Turing ideal generated by a chain of 2-random reals that contains no x-random real for any x in the ideal.

Background

Property (i) states that every ideal generated by a 2-random real contains a random real relative to each of its members. The question asks whether a chain-generated ideal can fail this property.

References

Is there a Turing ideal $B$ generated by a chain of $2$-random reals such that $B$ contains no $x$-random real for any $x\in B$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 1, Problem, item (3)