Is it true that the quantile function of an $L^1$ random variable is $L^2(]0,1[)$?

Let $F^{-1}:=Q$. It is well known that $Y:=F^{-1}(U)=Q(U)$ equals $X$ in distribution, where $U$ is any random variable uniformly distributed on $(0,1)$. Therefore, for any real $p>0$, $$\int_0^1|Q(u)|^p\,du=E|Y|^p=E|X|^p.$$ So, $Q\in L^p((0,1))$ iff $X\in L^p(\Omega,\mathcal A,P)$, for each real $p>0$.