Let $E^n$ be the real or complex space of dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if $A$ is an endomorphism, then $$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$ is an operator norm of $A$.

The Banach-Mazur distance between $N$ and $M$ is $$d(N,M):=\log\inf_{A\in Aut(E)}\|A\|^M_N\|A^{-1}\|^N_M.$$ It is actually a semi-distance, with $d(N,M)=0$ iff one passes from $N$ to $M$ by a change of coordinates. Thus $d$ is a distance on some quotient space, in which there is only one Euclidian norm, for instance.

Let us define the usual $\ell^p$-norm by $$\|x\|_p:=\left(\sum_j|x_j|^p\right)^{1/p}.$$ The $L^\infty$-norm is as usual. It has been known for a long time that if $1\le p,q\le2$, or if $2\le p,q\le\infty$, then $$d(\ell^p,\ell^q)=\left|\frac{1}{p}-\frac{1}{q}\right|\log n.$$ What is known if $1\le p\le2\le q\le+\infty$ ?