banach spaces – On weakly precompact sets

Recall that a bounded subset $A$ of a Banach space $X$ is said to be weakly precompact if every sequence in $A$ admits a weakly Cauchy subsequence. Rosenthal’s $l_{1}$-theorem states that a bounded subset $A$ is weakly precompact if and only if it contains no $l_{1}$-sequence. For a bounded subset $A$ of $X$, we let $$operatorname{wpc}_{X}(A)=inf{epsilon>0:Asubseteq K_{epsilon}+epsilon B_{X}},$$ where $K_{epsilon}$ is weakly precompact.

It is easy to see that $operatorname{wpc}_{X}(A)=0$ if and only if $A$ is weakly precompact. Clearly, every weakly Cauchy sequence is weakly precompact. Next I want to know how to quantify this simple fact.

Let $(x_{n})_{n}$ be a bounded sequence of $X$. We let $$delta((x_{n})_{n})=sup_{x^{*}in B_{X^{*}}}inf_{n}sup_{k,lgeq n}|langle x^{*},x_{k}-x_{l}rangle|.$$ Clearly, $delta((x_{n})_{n})=0$ if and only if $(x_{n})_{n}$ is weakly Cauchy. It is easy to see that $delta((x_{n})_{n})$ is the diameter of the set of all $weak^{*}$-cluster points of $(x_{n})_{n}$ in $X^{**}$.

Question. $operatorname{wpc}_{X}((x_{n})_{n})leq delta((x_{n})_{n})$

Thank you !