Alternative characterizations of asymptotic dimension for fuzzy metric spaces

Let $X$ be a fuzzy metric space, and let $n$ be a nonnegative integer. Conditions,, and in Theorem are the three conditions stated there for $X$ and $n$. Alternative-characterization conjecture. Conditions,, and are equivalent to

$$\operatorname{asdim}X \leq n.$$

Consequently, these conditions would provide alternative definitions of asymptotic dimension for fuzzy metric spaces. The supplied text gives no evidence that this assertion has been proved or disproved.