Given $n$, is there a $\big(1+C(n)\big)$-isomorphic embedding of $l^n_{\infty}$ into $l^m_1$ for sufficiently large $m$ and $C(n)<<\log(n)$?

For $n=2$ this can done with $m=2$.There are some results about $(1+\epsilon)$-isometric embedding of $l^n_p$ into $l^m_1$ for $p\leq 2$ but I couldn't find anything for $p>2$.