Estimation of a quadratic function and the $p$-Banach–Saks property

Let $E$ be a rearrangement-invariant Banach function space on $[0,1]$, and let $\Gamma(E)$ denote the set of all $p\ge 1$ such that any sequence $\{x_n\}$ in $E$ converging weakly to 0 has a subsequence $\{y_n\}$ with $\sup_m m^{-1/p}\|\sum_{1\le k\le m}y_n\|<\infty$. The set $\Gamma_i(E)$ is defined similarly, but only sequences $\{x_n\}$ of independent random variables are taken into account. It is proved (under the assumption $\Gamma(E)\ne\{1\}$) that if $\Gamma_i(E)\setminus\Gamma(E)\ne\varnothing$, then $\Gamma_i(E)\setminus\Gamma(E)=\{2\}$.