Equality of cut time and Maxwell time on the Cartan group

Let $C$ be the set of covectors parametrizing sub-Riemannian geodesics on the Cartan group. For $\lambda \in C$, let $t_{\operatorname{cut}}(\lambda)$ denote its cut time and let $t_{\operatorname{MAX}\nolimits}^1(\lambda)$ denote the first Maxwell time.

Cut-time equality conjecture. For any $\lambda \in C$

$$t_{\operatorname{cut}}(\lambda) = t_{\operatorname{MAX}\nolimits}^1(\lambda).$$

This conjecture concerns global optimality of sub-Riemannian geodesics on the Cartan group. The paper proves the lower bound that the first conjugate time is at least the first Maxwell time and notes that equality of cut and Maxwell times was conjectured previously; the equality remains unresolved here.