The free Banach lattices generated by ${\mathrm{\ell }}_p$ and $c_0$

We prove that, when $2<p<\mathrm{\infty }$, in the free Banach lattice generated by ${\mathrm{\ell }}_p$ (respectively by $c_0$), the absolute values of the canonical basis form an ${\mathrm{\ell }}_r$-sequence, where $\frac{1}{r}=\frac{1}{2}+\frac{1}{p}$ (respectively an ${\mathrm{\ell }}_2$-sequence). In particular, in any Banach lattice, the absolute values of any ${\mathrm{\ell }}_p$ sequence always have an upper ${\mathrm{\ell }}_r$-estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable ${\mathrm{\ell }}_p(\mathrm{\Gamma })$ for $2<p<\mathrm{\infty }$, as well as $c_0(\mathrm{\Gamma })$, are weakly compactly generated whereas this is not the case for $1\le p\le 2$.