Stabilizing isomorphisms from $\ell_p(\ell_2)$ into $L_p[0,1]$

Let $1<p\not=2<\infty$, $ε>0$ and let $T:\ell_p(\ell_2)\overset{into}{\rightarrow}L_p[0,1]$ be an isomorphism. Then there is a subspace $Y\subset \ell_p(\ell_2)$ $(1+ε)$-isomorphic to $\ell_p(\ell_2)$ such that: $T_{|Y}$ is an $(1+ε)$-isomorphism and $T(Y)$ is $K_p$-complemented in $L_p[0,1]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+ε)γ_p$ if $p>2$ and $K_p\le (1+ε)γ_{p/(p-1)}$ if $1<p<2$, where $γ_r$ is the $L_r$ norm of a standard Gaussian variable.

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