Complete latticeability in vector-valued sequence spaces

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 $({\mathrm{\ell }}_p{\widehat{\otimes }}_{\mathrm{\mid }\pi \mathrm{\mid }}E)\setminus ({\mathrm{\ell }}_p{\widehat{\otimes }}_{\pi }E)$, where $E$ is a Banach lattice and $1<p<\mathrm{\infty }$.