Rifford's curvature exponent–geodesic dimension conjecture for Carnot groups

Let $N_{\mathrm{CE}}$ be the curvature exponent of a Carnot group, namely the least $N$ for which the measure contraction property $\mathrm{MCP}(0,N)$ holds. Let $N_{\mathrm{GEO}}$ denote its geodesic dimension, defined by

$$\sup_{p \in \mathbb G}\left[ \inf\left\{ N > 0: \sup_{A \in \mathcal F} \limsup_{s \to 0} \frac{\mu(Z_s(p,A))}{s^N \mu(A)} = \infty \right\} \right],$$

where $\mathcal F:= \{A \subset \mathbb G \text{ bounded, measurable with }0< \mu(A) <\infty\}$. Rifford's curvature exponent–geodesic dimension conjecture. In the setting of Carnot groups with negligible cut loci, we have $N_{\mathrm{CE}} = N_{\mathrm{GEO}}$. The conjecture was disproved: there are step-two Carnot groups for which $N_{\mathrm{CE}} > N_{\mathrm{GEO}}$.