2-randoms with bounded complexity difference

Determine whether there exist 2-random reals x<_T z whose prefix-free complexity values differ by O(1) at every length.

Background

This asks for comparable 2-random reals with unusually close initial-segment complexity, another proposed route toward a negative answer to Levin’s representation problem.

References

Are there $2$-random reals $x<_T z$ such that $ \left| K(x\upharpoonright n)-K(z\upharpoonright n)

\right|

O(1)?$

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 1, Problem, item (2)