Let $a_1,\dots,a_n$ be real numbers such that $a_1^2+\dots+a_n^2=1$. Let $\eta_1,\dots,\eta_n$ be independent Rademacher random variables (r.v.'s), so that $P(\eta_i=\pm1)=\frac12$ for all $i$. Let $S:=a_1\eta_1+\dots+a_n\eta_n$, and let $T$ be an independent copy of $S$. Does then the inequality $$(*)\qquad E f_p(S)\le E f_p\Big(\frac{S+T}{\sqrt2}\Big)$$ hold for all real $p\ge2$, where $f_p(x):=|x|^p$?

This conjecture was communicated to me by Joel Zinn quite some time ago, and I think it deserves to be more broadly known. A motivation behind it was to obtain an alternative (and hopefully easier) proof of Haagerup's inequality $$E f_p(S)\le E f_p(Z)$$ Haagerup, which indeed easily follows from $(*)$ by the central limit theorem; here $Z$ is a standard normal r.v. For $p=2$, $(*)$ is trivial. For $p\ge3$, $(*)$ is easily proved; in particular, it follows immediately from Corollary 2.5 in T^2.

So, actually the question is only about $p\in(2,3)$. In that case, in view of Lemma 2.2 in Figiel et al, it would be enough to prove $(*)$ with $g_t$ in place of $f_p$ (for all real $t>0$), where $g_t(x):=|x|^3-\max(0,|x|-t)^3$.