Weakly null sequences in $L_{p}(1<p<2)$

Question

Let $1<p<2$. Let $(f_{n})_{n}$ be a normalized weakly null sequence in $L_{p}$ such that the sequence $(f_{n})_{n}$ contains no subsequence that is equivalent to the unit vector basis of $l_{p}$.

Question: Does $(f_{n})_{n}$ admit a subsequence $(f_{k_{n}})_{n}$ such that $$\|\sum_{n=1}^{m}a_{n}f_{k_{n}}\|_{p}\leq C_{p}(\sum_{n=1}^{m}|a_{n}|^{2})^{\frac{1}{2}},$$ for all $m\in \mathbb{N}$ and all scalars $a_{1},a_{2},...,a_{m}$. The constant $C_{p}$ depends only on $p$.

By Orlicz's Theorem, the reverse of the above inequality is true.

Thank you!

Answer 1

This question, as stated, has an easy negative answer: $L_p$, $1<p<2$ contains a weakly null sequence equivalent to the unit vector basis of $\ell_p$, and so does not satisfy the mentioned condition.