Let $S$ be a compact orientable 2-surface with $\chi(X)\leq -3$. Y.Minsky and H.Masur proved that the curve complex of $X$ is $\delta$-hyperbolic and infinite. E.Klarreich (see also U.Hamenstadt) proved the Gromov boundary of the curve complex of $S$ is bijective to the collection of ending laminations. Denote the collection of ending laminations by $B$.

Note: ending lamination implies that its complement in $S$ is a collection of (once-punctured) ideal polygons.

My question is: Given a number $N$, is there possible that there is a collection of essential simple closed curves $Y=\{c_{\eta} \mid \eta\in B\}$ such that $d_{\mathcal {C}(S)}(c_{\eta}, c_{\zeta})\geq N$, for any $\eta, \zeta \in B$ and $\eta \neq \zeta$?