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 $\lbrace x_n \rbrace$ in $E$ converging weakly to 0 has a subsequence $\lbrace y_n \rbrace$ with $\sup_m m^{-1/p}\Vert\sum_{1\le k\le m} y_n\Vert<\infty$. The set $\Gamma_i(E)$ is defined similarly, but only sequences $\lbrace x_n \rbrace$ of independent random variables are taken into account. It is proved (under the assumption $\Gamma(E)\ne \lbrace1\rbrace$) that if $\Gamma_i(E)\setminus\Gamma(E)\ne\varnothing$, then $\Gamma_i(E)\setminus\Gamma(E)=\lbrace 2\rbrace$.