Optimal curved Brunn–Minkowski inequality for the symmetric group

Let $S(n)$ be the symmetric group equipped with the metric $\mathrm{d}$, and let $A,B\subseteq S(n)$ be nonempty. Write $M$ for their midpoint set, and let $\mathrm{d}(A,B)$ denote the distance between the sets. Set

$$K=\frac{c}{n-1},$$

where $c$ is a positive constant. Optimal curved Brunn–Minkowski conjecture. For any nonempty $A,B\subseteq S(n)$,

$$\log |M|\geq \frac{1}{2}\log|A|+\frac{1}{2}\log|B|+\frac{K}{8}\mathrm{d}(A,B)^2.$$

This is proposed as an optimal curvature strengthening of the curved Brunn–Minkowski inequality proved earlier in the paper. The source states that it is currently unproved and that it follows from the conjectural concentration inequality given separately below.