Let $E$ be a Banach space, let $e_{n}\in E$ and $g_{n}\in E^{*}$ be biorthogonal basic sequences (i.e. $\left<e_n,g_m\right>=\delta_{mn}$ ). Moreover, both of these sequences are weakly null. (note that existence of these sequences is equivalent to negation of Dunford-Pettis property)

Can we always find $\alpha>0$, an infinitely dimensional subspace $H\subset E$ and a weakly compact $D\subset E^{*}$, such that $\sup\limits_{d\in D} |\left<h,d\right>|\ge \alpha \|h\|$, for every $h\in H$?

(note that this implies reflexivity of $H$)