Let $1<p<\infty$. Recall that a Banach space $X$ is $l_{p}$-saturated if every infinite-dimensional subspace of $X$ contains a subspace isomorphic to $l_{p}$. I have a seemingly stronger notion of $l_{p}$-saturated Banach spaces. We say that a Banach space $X$ is strongly $l_{p}$-saturated if for every infinite-dimensional subspace $M$ and every $\epsilon>0$, there exists an infinite-dimensional subspace $N$ of $M$ such that $N$ is $(1+\epsilon)$-isomorphic to $l_{p}$. I have the following two questions:

Question 1. If a Banach space $X$ is $l_{p}$-saturated, is $X$ strongly $l_{p}$-saturated?

Question 2. Are there other examples of strongly $l_{p}$-saturated spaces besides $l_{p}$? Is $L_{p}$ strongly $l_{p}$-saturated?

Thank you!