The complex Banach–Mazur distance conjecture for ℓ₁ and ℓ∞

For $n\geq 1$, let $\ell_1^n(\mathbb{C})$ and $\ell_{\infty}^n(\mathbb{C})$ denote the $n$-dimensional complex sequence spaces, and let $d_{BM}$ denote the Banach–Mazur distance between normed spaces. The complex Banach–Mazur distance conjecture. For every integer $n\geq 1$,

$$d_{BM}\bigl(\ell_1^n(\mathbb{C}),\ell_{\infty}^n(\mathbb{C})\bigr)=\sqrt{n}.$$

The paper notes that the upper bound is known in every dimension, while the lower bound would follow from the unimodular Spencer conjecture. The exact value was stated to be unpublished even for $n=2$, so the conjecture remains open in the supplied text.