Non-degeneracy conjecture for the α-metric on closed connected Legendrian submanifolds

Let $(M,\xi=\ker\alpha)$ be a contact manifold with fixed contact form $\alpha$, and let $N$ be a closed connected Legendrian submanifold. Let $\mathcal L(N)$ be the orbit space of $N$ under ${\rm Cont}_0(M,\xi)$, and for $L_1,L_2\in\mathcal L(N)$ define

$$\delta_{\alpha}(L_1,L_2)=\inf\{\lVert\phi\rVert_{\alpha}\mid \phi\in {\rm Cont}_0(M,\xi)\text{ and }\phi(L_1)=L_2\}.$$

Non-degeneracy conjecture. The $\alpha$-metric $\delta_{\alpha}$ on $\mathcal L(N)$ is non-degenerate.

The paper’s main dichotomy theorem shows that for a closed connected submanifold of dimension $n$ in a contact manifold of dimension $2n+1$, the $\alpha$-metric is either non-degenerate or vanishes identically. The stated Legendrian claim asserts the non-degenerate alternative, but the supplied text does not establish its general validity or provide a resolution.