A Banach space $X$ is said to have the Dunford-Pettis property if for any Banach space $Y$ every weakly compact operator $T:X\rightarrow Y$ is completely continuous. Recall that $T$ is completely continuous if it maps weakly convergent sequences to norm convergent sequences, or equivalently, if it maps weakly Cauchy sequences to norm Cauchy ones. There are many well-known characterizations of the Dunford-Pettis property. Recently I am thinking about the following characterization:

Let $X$ be a Banach space. The following are equivalent:

(1) $X$ has the Dunford-Pettis property.

(2) There is $C>0$ so that $\limsup\limits_{n}|\langle x^{*}_{n},x_{n}\rangle|\leq C\delta((x_{n})_{n})$ whenever $(x_{n})_{n}$ is a bounded sequence in $X$ and $(x^{*}_{n})_{n}$ is a weakly null sequence in $B_{X^*}$, where $\delta((x_{n})_{n})=\sup\limits_{x^{*}\in B_{X^{*}}}\inf\limits_{n}\sup\limits_{k,l\geq n}|\langle x^{*},x_{k}-x_{l}\rangle|$.

I am not sure if this characterization is true. Thank you!