Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]).

I am wondering whether are any result for the case when $r>s>2$?

[Johnson, William B.; Schechtman, Gideon Embedding $l_p^m$ into $l_1^m$, Acta Math. 149 (1982), 71--85.][JS]