Determine the asymptotic Banach–Mazur radius of \(\ell_\infty^n\)

Determine whether the upper bound \(R_{BM}(\ell_\infty^n)\lesssim n^{2/3}\) is optimal up to lower-order factors by establishing or refuting the corresponding matching lower bound for the Banach–Mazur radius of \(\ell_\infty^n\).

Background

The paper proves the upper bound $R_{BM}(\ell_\infty^n)\lesssim n^{2/3}$, while the best lower bound cited in the introduction is of order $n^{5/8}$ up to polylogarithmic factors. The authors note that the exponent $2/3$ has an intrinsic meaning for both the known lower-bound examples and their proof strategy, suggesting—but not establishing—that the new upper bound may be optimal up to lower-order factors. Because the authors explicitly state that turning this suspicion into a conjecture would be speculative, the optimal asymptotic growth remains unresolved.