Bounded number of stable sets for n-expansive homeomorphisms

Let $f$ be an $n$-expansive homeomorphism of a compact metric space $X$. For $\bar{\varepsilon}>0$ and $x\in X$, let $n(x,\bar{\varepsilon})$ denote the number of different stable sets of $f$ in $W^s_{\bar{\varepsilon}}(x)$, namely the largest positive integer satisfying the two conditions that there is a set of that cardinality whose distinct points lie in distinct stable sets, and that every set of one larger cardinality contains two points in the same stable set.

Bounded stable-set conjecture. There exists $\varepsilon>0$ such that

$$n(x,\bar{\varepsilon})\leq n$$

for every $x\in X$ and every $0<\bar{\varepsilon}\leq\varepsilon$.

The conjecture seeks a uniform bound, determined by the expansiveness constant $n$, on the number of distinct stable sets occurring in sufficiently small local stable sets. The supplied text does not state whether this conjecture has been resolved.