By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at a {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many points on matrices of order $3$ \begin{bmatrix} 1 & 1 & 1\\ -1 & 1 & 1\\ t & 1 & -1 \end{bmatrix} $-1 \leq t \leq 1$. However, I verified that there are only finitely many maximum points for orders $4, 5, 6$. So, I ask, is the maximum attained only at finitely many points for order above $3$, or perhaps for orders not equal to $3$ mod 4? I suspect this is known, as there has been a lot of work on Hadamard Maximum determinant problem, but I can't find such statement so far. If this is not known, can we at least prove that the dimension of the space of maximum points is less than the order?