  We prove the super-exponential decay of probabilities that there exist $n$
crossings of a given quadrilateral in a simple $\text{CLE}_\kappa(\Omega)$,
$\frac{8}{3}<\kappa\le 4$, as $n$ goes to infinity. Besides being of
independent interest, this also provides the missing ingredient in
arXiv:1809.00690 for proving the convergence of probabilities of cylindrical
events for the double-dimer loop ensembles to those for the nested
$\text{CLE}_4(\Omega)$.
