Equality of particle-system and mean-field log-Sobolev constants

Let $\lambda^N_{\operatorname{LS}}$ denote the log-Sobolev constant of the $N$-particle system, and let $\lambda^\infty_{\operatorname{LS}}$ denote the corresponding mean-field constant. Assume that the hypotheses A1 and A2 hold.

Log-Sobolev constant limit conjecture. Under assumptions A1 and A2,

$$\lim_{N\to\infty}\lambda^N_{\operatorname{LS}}=\lambda^\infty_{\operatorname{LS}}.$$

This conjecture would complete the characterisation of the limiting log-Sobolev constant in terms of the mean-field system. The paper gives evidence through degeneracy and contraction results, but states that the full characterisation remains unresolved.