Positive absolute lower bound for the normalized symmetric KSZ constant

Establish whether there exists a positive absolute constant c such that C_m^{sym}/\sqrt{m!} \ge c for all sufficiently large degrees m, where C_m^{sym} is the dimension-free real symmetric Kahane–Salem–Zygmund constant defined by the least normalized supremum norm of real symmetric unimodular m-linear forms on (\ell_\infty^n)^m.

Background

The paper studies the dimension-free constant C_m^{sym} governing the smallest supremum norm achievable by real symmetric unimodular m-linear forms on (\ell_\infty^n)^m. The authors prove an upper bound C_m^{sym} \le C_0\sqrt{m!} with an absolute constant C_0, while their Walsh-spectrum argument gives the asymptotic lower bound C_m^{sym} \ge (\sqrt{2/e}+o(1))\,m\sqrt{m!}.

These estimates determine the factorial scale but leave a multiplicative gap of order m between the available lower and upper bounds. The unresolved issue is whether the normalized quantity C_m^{sym}/\sqrt{m!} stays uniformly positive, which would improve the lower bound to the same factorial order as the upper bound.