We say that a sequence $(x_{n})_{n}$ in a Banach space $X$ is unconditionally $p$-summable ($1\leq p<\infty$) if $$\sup_{x^{*}\in B_{X^{*}}}(\sum_{n=m}^{\infty}|\langle x^{*},x_{n}\rangle|^{p})^{\frac{1}{p}}\rightarrow 0\quad (m\rightarrow \infty).$$ We denote the set of all unconditionally $p$-summable sequences on $X$ by $l^{u}_{p}(X)$. Define a norm $$\|(x_{n})_{n}\|_{p}^{w}=\sup\{(\sum_{n=1}^{\infty}|\langle x^{*},x_{n}\rangle|^{p})^{\frac{1}{p}}:x^{*}\in B_{X^{*}}\}, (x_{n})_{n}\in l^{u}_{p}(X).$$
The following is my question: Let $(x_{n})_{n}\in l^{u}_{p}(X)$ and let $\epsilon>0$. Are there $(\widetilde{x_{n}})_{n}\in l^{u}_{p}(X)$ and $(\lambda_{n})_{n}\in c_{0}$ such that $\|(\lambda_{n})_{n}\|=1$, $x_{n}=\lambda_{n}\widetilde{x_{n}}$ ($n=1,2,...$) and $\|(\widetilde{x_{n}})_{n}\|_{p}^{w}<\|(x_{n})_{n}\|_{p}^{w}+\epsilon$?
