Completely symmetric subspaces of noncommutative $L_p$-spaces

Let $2<p<\mathrm{\infty }$. We prove that every infinite-dimensional closed subspace $F$ of a noncommutative $L_p$-space contains a subspace completely isomorphic to ${\mathrm{\ell }}_p$, $C_p$, $R_p$, or $C_p\cap R_p$ and completely complemented in $F$. This answers the question of Junge and Xu on the existence of completely symmetric basic sequences.