The Deneanu–Vu conjecture on normal sign matrices

A real matrix is normal if it commutes with its adjoint; for real matrices this means

$$MM^{T}=M^{T}M.$$

Let a sign matrix be a matrix whose entries belong to $\{\pm1\}$. Since every real symmetric sign matrix is normal, there are at least $2^{\binom{n+1}{2}}$ $n\times n$ normal sign matrices. Deneanu–Vu conjecture. There are

$$2^{(0.5+o(1))n^2}$$

$n\times n$ $\{\pm1\}$-valued normal matrices. This asserts that the symmetric-matrix lower bound is essentially sharp; the source attributes the conjecture to Deneanu and Vu, and no resolution is supplied.