LR-degree intermediate-value question

Determine whether the intermediate-degree property for Martin-Löf randomness holds when Turing degrees are replaced by LR-degrees.

Background

The analogous assertion for Turing degrees is stated to have a negative answer. The question asks whether a stronger degree notion restores the property.

References

Does this hold with a larger degree notion, for example, LR-degrees?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 9