Half-factor conjecture for full-domain permutation groups

Prove or refute that every finite permutation group G acting on its full domain Ω satisfies dimN(H_{G,Ω}) ≥ ⌈H(G)/2⌉, where dimN is the Natarajan dimension and H(G) is the permutation-group height.

Background

The paper establishes the inequality when the closure operator induced by pointwise stabilizers is a matroid closure, and shows that the constant 1/2 is sharp for symmetric groups and products such as S_3^k. The conjecture concerns arbitrary finite permutation groups acting on the entire underlying set, including groups whose closure operators are not matroidal.

The authors note that commuting stabilizer moves do not suffice in general, so a proof may need to construct Natarajan squares from noncommuting moves. A counterexample would refute the proposed universal lower bound.