Let $X$ be a Banach space. We let $j=e^{2i\pi/3}$. Let $(\epsilon_i)$ be a sequence of independent Rademacher variables on a fixed probability space $\Omega$. Let $(\varphi_i)$ be a sequence of independent complex random variables such that $P(\varphi_i=1)=\frac{1}{3}$, $P(\varphi_i=j)=\frac{1}{3}$ and $P(\varphi_i=j^2)=\frac{1}{3}$ for any $i$.
Do there exist sufficient conditions on $X$ such that there exist constants $m,M>0$ such that $ m\Big\vert\Big\vert\sum_i \epsilon_i\otimes x_i \Big\vert\Big\vert_{L^p(\Omega,X)}\leq \Big\vert\Big\vert\sum_i \varphi_i \otimes x_i \Big\vert\Big\vert_{L^p(\Omega,X)}\leq M\Big\vert\Big\vert\sum_i \epsilon_i\otimes x_i \Big\vert\Big\vert_{L^p(\Omega,X)} $ for any $x_1,\ldots,x_n\in X$?
Remark: I know that if $X$ has finite cotype, the Rademacher averages and the gaussian averages are equivalent.