Inspired by this very related question.
Let $A$ be a circulant matrix and $X$ an anti-circulant matrix of size $n\times n$. Moreover let the sum of each row in $X$ be zero, $\sum_i x_{ij}=0$. It appears that for $n=3$, Oppenheim's inequality achieves equality, i.e.: $$\det(A\circ X)= (\det A)\prod_i x_{ii}$$
Where $\circ$ is the Hadamard product.
Empirically, this relation no longer holds for other $n$. Is there a particular reason why this only works for $n=3$ and not for higher $n$? Is this just chance or is there a deeper reason?