  We consider a random walk in an i.i.d. Cauchy-tailed conductances
environment. We obtain a quenched functional CLT for the suitably rescaled
random walk, and, as a key step in the arguments, we improve the local limit
theorem for $p^{\omega}_{n^2t}(0,y)$ in [Ann. Probab. (2009). To appear],
Theorem 5.14, to a result which gives uniform convergence for
$p^{\omega}_{n^2t}(x,y)$ for all $x,y$ in a ball.
