The conjecture that the cut time equals the first coordinate cut time on Grushin spaces

Let $x(t;\phi)$ be a non-trivial geodesic in the Grushin space $\mathbb{G}^{n+1}_\alpha$. For $j\in J$, let $\tau_j$ be the quantities defined by the equation referenced in the source, and set

$$\tau=\min_{j\in J}\tau_j.$$

Cut-time conjecture. The cut time $t_{\operatorname{cut}}$ is identically

$$t_{\operatorname{cut}}=\tau=\min_{j\in J}\tau_j.$$

The preceding theorem proves only the upper bound $0\leqslant t_{\operatorname{cut}}\leqslant\tau$; the conjecture asserts that no third geodesic intersects earlier, so the bound is always sharp.