Existence of logarithmic-ratio attaining sequences for compact metric spaces

Let $(X,d)$ be a compact metric space, and let $R(X,d)$ denote its logarithmic ratio. For $p\in(1,\infty)$, let ${\mathcal A}^{\ast}_p(X)$ be the associated class of sequences.

Conjecture 2.8. If

$$0<R(X,d)<\infty,$$

then

$${\mathcal A}^{\ast}_p(X)\ne\emptyset$$

for some $p\in(1,\infty)$.

This conjecture would, together with the preceding results, imply that every compact metric space with finite positive logarithmic ratio is bi-Hölder equivalent to an ultrametric space and hence belongs to the universal class described via inverse limits of finite discrete spaces. The supplied context does not indicate whether the conjecture has been resolved.