Separability of the alternative $p$-variation topology

Determine whether the metric space of reduced weakly geometric $p$-rough paths equipped with the alternative metric $\delta^p_\star$ is separable, or in particular whether it is Polish, for $p>1$.

Background

The paper discusses several topologies on the reduced path group. The alternative metric $\delta^p_\star$ yields a Polish space when $p=1$, whereas the tree metric $\delta^p$ is known to produce a non-separable space. The authors express an unresolved expectation that the analogous alternative topology will fail to be separable for higher variation exponents because Hölder spaces are generally non-separable.