The weak maximizing property is not inherited by subspaces of the domain

We construct a separable reflexive Banach lattice $X$ and a closed sublattice $E\subseteq X$ such that every bounded linear operator from $X$ into ${\mathrm{\ell }}_2$ is compact, whereas there exists a positive operator from $E$ into ${\mathrm{\ell }}_2$ that does not attain its norm and admits a positive maximizing sequence converging weakly to a nonzero vector. This answers negatively the question posed by Dantas, Jung and Martínez-Cervantes in [4] concerning the inheritance of the weak maximizing property by closed subspaces of the domain. It also answers negatively the corresponding question for the positive weak maximizing property and closed sublattices of reflexive Banach lattices. Canonical lattice complexification yields a counterexample to the original question over the complex field as well.