In 1968, J. Lindenstrauss and A. Pe{\l}czy'{n}ski posed a problem: Is every complemented subspace $X$ of an $\mathcal{L}_{p}$-space ($1\leq p\leq \infty$) either an $\mathcal{L}_{p}$-space or isomorphic to a Hilbert space? If $p=1$ or $\infty$ and $X$ is infinite-dimensional, $X$ can not be a Hilbert space. In 1969, J. Lindenstrauss and H. P. Rosenthal stated that

Theorem. If $Y$ is an $\mathcal{L}_{p,1+\epsilon}$-space for every $\epsilon>0$ and $X$ is a $1$-complemented subspace of $Y$, then $X$ is an $\mathcal{L}_{p,1+\epsilon}$-space for every $\epsilon>0$.

They mentioned that the proof of the theorem can be seen in J. Lindenstrauss and A. Pe{\l}czy'{n}ski's paper. However, I can not see any of its proof.

Question 1. In what reference is there a proof of Theorem ?

Question 2. If $X$ is a $C$-complemented subspace of $Y$ in Theorem, is $X$ an $\mathcal{L}_{p,\lambda}$-space for some $\lambda$ depending only on $C$ ?

Thank you!