The complex cross-2 Banach–Mazur distance conjecture

For $n\geq 1$, let $\ell_p^n(\mathbb{C})$ denote the $n$-dimensional complex sequence space with exponent $p$, and let $d_{BM}$ denote the Banach–Mazur distance. For $1\leq q\leq 2\leq p\leq\infty$, define

$$\alpha=\max\left\{\frac{1}{2}-\frac{1}{p},\frac{1}{q}-\frac{1}{2}\right\}.$$

The complex cross-2 distance conjecture. For every integer $n\geq 1$ and all $1\leq q\leq 2\leq p\leq\infty$,

$$d_{BM}\bigl(\ell_p^n(\mathbb{C}),\ell_q^n(\mathbb{C})\bigr)=n^{\alpha}.$$

This extends the proposed exact formula for the complex distance between $\ell_1^n$ and $\ell_{\infty}^n$ to exponents on opposite sides of $2$. The source gives the corresponding order estimates and says that exact values in this regime are generally unknown; no resolution is supplied.