The nonintegral curvature exponent conjecture for step-two Carnot groups

Let $N_{\mathrm{CE}}$ denote the curvature exponent of a step-two Carnot group, namely the least $N$ for which the measure contraction property $\mathrm{MCP}(0,N)$ holds. Nonintegral curvature exponent conjecture. There exists a step-two Carnot group on which $N_{\mathrm{CE}}$ is not an integer. The paper motivates this conjecture by conjecturing continuity of $N_{\mathrm{CE}}$ under the relevant Lie algebra convergence, while noting that the cut times of all geodesics on step-two Carnot groups are not known.