Let $p>2$ and $X$ a subspace of $L_{p}$.

Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$.

Question: if $X$ is isomorphic to $l_{2}$, does $X$ contain a subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$?