Let $p>2$. Following from M.I. Kadec and A. Pełczyński's results (Studia Math. 1962), R.J.Whitley (Trans. Amer. Math.Soc. 1964) observed that $L_{p}$ is subprojective, that is, every infinite-dimensional closed subspace of $L_{p}$ contains an infinite-dimensional closed subspace that is complemented in $L_{p}$. What I am thinking about is the universal subprojectivity of $L_{p}$. Namely, I have the following question:

Question: Does every infinite-dimensional closed subspace of $L_{p}$ contain an infinite-dimensional closed subspace that is $C_{p}$-complemented in $L_{p}$? where the constant $C_{p}$ depends only on $p$.

Let $E$ be an infinite-dimensional closed subspace of $L_{p}$. If $E$ is not isomorphic to $l_{2}$, then an observation of the proof of Theorem 2 in M.I. Kadec and A. Pełczyński's paper (Studia Math. 1962) shows that $E$ contains an infinite-dimensional closed subspace that is $\lambda_{p}$-complemented in $L_{p}$, where the constant $\lambda_{p}$ depends only on $p$. But, I do not know whether this is true for the case $E$ is isomorphic to $l_{2}$.

Thank you!