Let $p>2$. Let $X$ be a subspace of $L_{p}$. Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$. My question is: if $X$ is isomorphic to $l_{2}$, does $X$ contain a subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$?

Thank you!