Uniform concentration inequality for symmetric log-concave measures

A symmetric log-concave probability measure is a symmetric probability measure whose density is log-concave. For a constant $C$, write $\mathrm{CI}(C)$ for the concentration inequality defined in the paper.

Uniform concentration inequality conjecture. Any symmetric log-concave probability measure satisfies $\mathrm{CI}(C)$ for some universal constant $C$.

The paper explains that this concentration conjecture follows from the corresponding infimum convolution conjecture, and that the two would be equivalent under the Kannan–Lovász–Simonovits conjecture.