Curvature exponent five characterization for sub-Finsler Heisenberg groups

Let $(\mathbb{H},\mathsf d)$ be a sub-Finsler Heisenberg group associated with a $C^2$ and strongly convex reference norm, and let $\mathscr{L}^3$ denote Lebesgue measure. The metric measure space $(\mathbb{H},\mathsf d,\mathscr{L}^3)$ satisfies $\mathsf{MCP}(0,5)$.

Curvature exponent five conjecture. The metric measure space $(\mathbb{H},\mathsf d,\mathscr{L}^3)$ satisfies $\mathsf{MCP}(0,5)$ if and only if the reference norm is the $\ell^2$-norm; equivalently, $(\mathbb{H},\mathsf d)$ is the sub-Riemannian Heisenberg group.

The curvature exponent is known to be at least $5$ for $C^2$ and strongly convex reference norms, while examples show that the lower bound can be strict. The conjecture asserts that equality characterizes the sub-Riemannian case.