Convex order under majorization for projections of ell-p balls

Let $n\ge1$ and $1\le p\le2$, and let $X$ be uniformly distributed on the unit ball $B_p^n$ of $\ell_p^n$. We prove that if $(a_1^2,\ldots,a_n^2)$ majorizes $(b_1^2,\ldots,b_n^2)$, then $\langle b,X\rangle^2$ is dominated by $\langle a,X\rangle^2$ in convex order. For $p<2$ this gives a nontrivial c