Rigidity conjecture for conjugation-invariant Lorentzian distance functions

Let $(M,xi)$ be a closed contact manifold and let $\mathrm{Cont}_0(M)$ denote the identity component of its contactomorphism group. Write $\llcurly$ for the chronological relation and $\preccurlyeq$ for its causal analogue. Let

$$\tau\colon \mathrm{Cont}_0(M)\times\mathrm{Cont}_0(M)\rightarrow [0,\infty]$$

be a map satisfying $\tau(\phi,\psi)>0$ if and only if $\phi\llcurly\psi$, and

$$\tau(\phi_1,\phi_2)\geq \tau(\phi_1,\psi)+\tau(\psi,\phi_2)$$

whenever $\phi_1\preccurlyeq\psi\preccurlyeq\phi_2$. Assume that $\tau$ is lower semicontinuous with respect to the interval topology and is conjugation invariant. Rigidity conjecture. Then

$$\tau(\phi,\psi)=\begin{cases}\infty,&\text{if }\phi\llcurly\psi,\\0,&\text{otherwise.}\end{cases}$$

This asserts that the stated positivity, reverse-triangle inequality, lower semicontinuity, and conjugation invariance force the only possible distance to be the trivial infinite-valued chronological distance. The source does not provide evidence resolving this claim, so its status remains open.