Latency-independent polynomial-time approximation exponent

Determine whether, for finite latency L, there exists a polynomial-time algorithm that approximates the Young domination number γ^L within a constant factor using a polynomial running-time exponent independent of L.

Background

For each fixed finite L, the paper obtains a constant-factor approximation algorithm, but the exponent in its polynomial running time depends on L. The authors explicitly ask whether the dependence of that exponent on latency can be eliminated while retaining a constant-factor approximation for finite L.

References

For finite $L$, is there an algorithm that approximates $\gammaL$ up to a constant factor in polynomial time with powers independent of $L$?

— Young domination on Hamming rectangles  (2501.03788 - Gravner et al., 7 Jan 2025) in Section 1, item 2 of Section 7 ("Open problems and possible further directions")