Conjectured implication from gradient-flow relaxation to uniform log-Sobolev inequality

Ascertain whether a uniform rate of exponential relaxation for the gradient flow associated with the free energy functional \mathcal F_T implies a uniform-in-N log-Sobolev inequality for the mean-field particle system measure m^N_T, as conjectured in Conjecture 1 of [MR4604897].

Background

Beyond the strong convexity criterion on \hat{\mathcal F}_T, the authors point to a broader question of linking dynamical relaxation properties of the gradient flow of \mathcal F_T to functional inequalities for the particle system, inspired by the McKean–Vlasov framework (in the sense of [MR2053570]).

They reference a precise conjecture in [MR4604897] asserting that uniform exponential decay of the gradient-flow energy could entail uniform log-Sobolev inequalities for the interacting particle system, and highlight this direction as a key open problem.