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Are all PLS-complete problems μ-downward self-reducible?

Determine whether every PLS-complete problem admits a μ-downward self-reduction.

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Background

The paper shows that, unless PLS = FP, not all PLS-complete problems are traditionally downward self-reducible, via a padding argument. This leaves open whether the generalized μ-d.s.r notion could still capture all PLS-complete problems.

Resolving this would clarify the completeness of μ-d.s.r as a unifying framework for PLS.

References

We have therefore shown that it is unlikely that every \cc{PLS}-complete problem is traditionally d.s.r. We leave open the question of whether every \cc{PLS}-complete problem is $\mu$-d.s.r.

— Downward self-reducibility in the total function polynomial hierarchy (2507.19108 - Gajulapalli et al., 25 Jul 2025) in Limitations of the d.s.r framework, Subsection: Not all PLS-complete problems are traditionally d.s.r