Approximate ultrahomogeneity in $L_p{L}_q$ lattices

We show that for $1\le p,q<\mathrm{\infty }$ with $p\mathrm{/}q\mathrm{\notin }\mathbb{N}$, the doubly atomless separable $L_p{L}_q$ Banach lattice $L_p(L_q)$ is approximately ultrahomogeneous (AUH) over the class of its finitely generated sublattices. The above is not true when $p\mathrm{/}q\in \mathbb{N}$. However, for any $p\mathrm{\ne }q$, $L_p(L_q)$ is AUH over the finitely generated lattices in the class $B{L}_p{L}_q$ of bands of $L_p{L}_q$ lattices.