Normal sequences without strongly normal subsequences

A sequence $(y_n^*)$ in the unit sphere of $E^*$ is strongly normal if the vectors $x\in E$ satisfying $\sum_n|y_n^*(x)|<\infty$ form a dense subspace. For every set $Γ$ of cardinality at least the continuum, we construct a normalized weakly null sequence in $\ell_1(Γ)^*$ with no strongly normal sub