Uniqueness of constant words as extremizers

Determine whether the constant word 0^m, together with its reversal and complementation variants, is the unique maximizer of qc among all binary words of length m for all sufficiently large m.

Background

The paper proves that qc(0m)=ceil(sqrt(m+1)), attaining the general upper bound for words of length m. Thus constant words are known to be extremal for their length.

It remains unresolved whether these constant-word symmetries are the only words attaining the maximum quantum Černý complexity when the word length is sufficiently large.

References

Is $0m$ (with its reversal and complementations) the unique maximizer of $qc$ among words of length $m$, for all large $m$?

— Quantum Černý complexity of binary words  (2609.40154 - Lee et al., 30 Sep 2026) in Section Discussion and open problems, subsection “Open problems,” item 4 (Extremal words)