  First we adjust a technique due to Jim\'enez-Rodr\'iguez to prove the
complete latticeability of the set of disjoint non-norm null weakly null
sequences and of the set of disjoint non-norm null regular-polynomially null
sequences in Banach lattices. Then we apply the mother vector technique to
prove the complete latticeability of $\lambda_\pi(E) \setminus \lambda_s(E)$,
which implies the complete latticeability of
$(\ell_p\widehat{\otimes}_{|\pi|}E)\setminus(\ell_p\widehat{\otimes}_{\pi}E)$,
where $E$ is a Banach lattice and $1 < p < \infty$.
