  We prove that, when $2<p<\infty$, in the free Banach lattice generated by
$\ell_p$ (respectively by $c_0$), the absolute values of the canonical basis
form an $\ell_r$-sequence, where $\frac{1}{r} = \frac{1}{2} + \frac{1}{p}$
(respectively an $\ell_2$-sequence). In particular, in any Banach lattice, the
absolute values of any $\ell_p$ sequence always have an upper
$\ell_r$-estimate. Quite surprisingly, this implies that the free Banach
lattices generated by the nonseparable $\ell_p(\Gamma)$ for $2<p<\infty$, as
well as $c_0(\Gamma)$, are weakly compactly generated whereas this is not the
case for $1\leq p\leq 2$.
