Uniform Kadec-Klee Property in Banach lattices

We prove that a Banach lattice $X$ which does not contain the $l^n_{\infty}$-uniformly has an equivalent norm which is uniformly Kadec-Klee for a natural topology $\tau$ on $X$. In case the Banach lattice is purely atomic, the topology $\tau$ is the coordinatewise convergence topology.