Hey my fellow Banach space guys. Sorry for another elementary question, and yes I have looked for the past hour and a half to see if I can find it on Google or in my books at home.

Fix $n\in\mathbb{Z}^+$. It seems like there should exist a strictly increasing sequence of $(p_k)\in[1,2)$ such that $p_k\to 2$ and $d(\ell_{p_k}^n,\ell_2^n)\to 1$, where

$$d(X,Y)=\inf\{\|T\|\|T^{-1}\|:T\in\mathcal{L}(X,Y)\text{ is an isomorphism}\}.$$

is the usual Banach-Mazur distance between isomorphic Banach spaces $X,Y$. Probably all I need is a rough estimate of the norm of the canonical isomorphism between $\ell_p^n$ and $\ell_2^n$ and the same for its inverse. But I am unable to show this.

Supposedly there is an estimate in some 1966 paper by Gurarii/Kadets/Matsaev, but it is in Russian. Also Tomczak-Jaegermann supposedly has mentioned the estimate in some of her papers/books, but they are unavailable from my university.

Any help would be much appreciated, thanks.