Failure of the measure contraction property with strong Goh–Legendre geodesics

Let $(M,\mathsf{d},\mathsf{m})$ be a sub-Riemannian metric measure space, and let a strong-Goh–Legendre geodesic be a geodesic satisfying the strong Goh and Legendre conditions. Write $\mathsf{d}^2$ for the squared distance function.

Failure of the measure contraction property with strong Goh–Legendre geodesics. If there is a strong-Goh–Legendre geodesic $\gamma$, then $\mathsf{d}^2:M\times M\to\mathbb{R}_+$ fails to be Lipschitz in charts in a neighbourhood of $\gamma$, and $(M,\mathsf{d},\mathsf{m})$ does not satisfy $\operatorname{MCP}(K,N)$ for all $K\in\mathbb{R}$ and $N\in[1,\infty)$.

This conjecture predicts that the presence of a strong-Goh–Legendre geodesic causes both local failure of Lipschitz regularity for the squared distance and failure of every measure contraction property. The surrounding discussion notes that the converse behaviour is known under the absence of Goh–Legendre geodesics, assuming real-analyticity, while the conjectured implication remains open.