TITLE: On sums of independent random variables in Banach spaces
QUESTION [14 upvotes]: Let $(\xi_n)_{n\ge 1}$, $(\eta_n)_{n\ge 1}$ be independent mean-zero random variables with values in a Banach space $X$ such that 
$$\sum_n\mathbb P(\xi_n\in A)\le\sum_n\mathbb P(\eta_n\in A)$$for any Borel set $A\subset X\setminus\{0\}$. 
Let $1\le p<\infty$. Is there a constant $C$ (perhaps depending on $p$ and $X$) such that
$$\textstyle\mathbb E\|\sum_n\xi_n\|^p\le C\,\mathbb E\|\sum_n\eta_n\|^p?$$

This problem was posed on 28.06.2019 on page 133 of Volume 2 of the Lviv Scottish Book.

REPLY [3 votes]: First of all, many thanks to @LvivScottishBook for putting the problem online (that was indeed very surprising for me) and to @fedja for finding a counterexample.
While writing the problem I was sure that it has a positive answer for any Banach space (for some reason this was kind of intuitive for me) with $C$ independent of $X$, so @fedja answered the original question. Nonetheless, as @fedja noticed, it is not clear for which Banach spaces such an inequality holds true. The only thing that I can guarantee so far is that the necessary assumption on the Banach space is having a finite cotype thanks to @fedja counterexample and the Maurey–Pisier theorem (see Corollary 7.3.14 in Analysis in Banach spaces, Volume II by Hytönen, van Neerven, Veraar, and Weis).
It remains open whether finite cotype is a sufficient condition as well.