Determinantal conjecture

Does the determinant of the sum $A + B$ of two $n \times n$ normal complex matrices $A$ and $B$ always lie in the convex hull of the $n!$ points $\prod_i (\lambda(A)_i + \lambda(B)_{\sigma(i)})$? Here the numbers $\lambda(A)_i$ and $\lambda(B)_i$ are the eigenvalues of $A$ and $B$, and $\sigma$ is an element of the symmetric group $S_n$.