Uniform bound on the DS-to-Natarajan ratio

Determine whether μ(Γ)/N(Γ) is bounded over all finite groups Γ, where μ(Γ) is the largest size of an independent subset and N(Γ) is the coset-cube number.

Background

For Galois classes, the DS dimension equals μ(Γ) while the Natarajan dimension equals N(Γ), so their ratio measures the possible loss in the agnostic ERM certificate. The paper gives examples with ratios greater than one, including A_5, S_5, and S_n for n≥5.

The half-factor conjecture would imply a universal bound of two. The authors also ask whether the stronger lower bound N(Γ)≥⌈2μ(Γ)/3⌉ holds; A_5 shows that the constant 2/3 could not be improved if that stronger statement is true.