The metric conjecture for the Teichmüller-type space of routes to chaos

For each route to chaos $C$, let $[C]$ be its boundary, namely the collection of limit dilatations of sequences drawn from the dilatation sets of the stages of $C$. Let $T_{chaos}$ be the collection of all such boundaries, and define

$$d([C],[C'])=\inf\{\ln(K(f^{-1}\circ g)): f\in[C],\ g\in[C']\}.$$

The function $d$ is already known in the paper to be symmetric and to satisfy the triangle inequality. Metric conjecture. $d$ is a metric on $T_{chaos}$. This strengthens the established pseudometric properties by asserting that distinct points have positive distance; the paper proposes a Teichmüller-theoretic approach but does not resolve the claim.