Approaching exponent one in the Erdős–Moser lower bound

Determine whether the exponent c in the lower bound φ(n)=Ω((log n)^{1+c}) obtainable through the k-configuration approach can be made arbitrarily close to 1.

Background

The paper proves that for every fixed c<1/68, sufficiently large finite sets of integers contain a subset of size at least (log |A|)^{1+c} that is sum-free with respect to the ambient set. It notes that the argument is intrinsically limited to c<1 in its present form and that optimization might improve the exponent. However, the authors explicitly leave unresolved whether the exponent can be pushed arbitrarily close to one, citing obstacles analogous to those in improving the Kelley–Meka bound for Roth’s theorem.