For this question, all Banach spaces are over the reals.

Let $1\leq p<\infty$. Recall that a sequence $(x_n)$ in a Banach space $E$ is weakly $p$-summable if $$ \Vert (x_n) \Vert_{p,w} := \sup_{\gamma\in E^* \colon \Vert\gamma\Vert\leq 1} \left( \sum_{n=1}^\infty \vert\gamma(x_n) |^p \right)^{1/p} < \infty .$$

Another way to think about this: for a Banach space $X$, bounded linear maps $X\to\ell_p$ correspond (isometrically) to weakly $p$-summable sequences in $X^*$.

Now suppose $1<p<2$ and let $q$ be the conjugate index of $p$.

Question.Let $(x_n)$ be a weakly $p$-summable sequence in $L_q[0,1]$. Is the sequence $(|x_n|)$ also weakly $p$-summable?

I suspect the answer is negative, but only because I've not had any luck finding a "soft" proof of a positive answer. On the other hand, the question seems natural enough that it must be in the literature one way or the other.