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 point 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?