A further sufficient condition for the determinantal conjecture

Let $A$, $B$ be $n\times n$ normal matrices with eigenvalues $(a_1,\ldots,a_n)$, $(b_1,\ldots,b_n)$, respectively. We show that $\det(A+B)$ lies in the convex hull of
$$\bigcup\limits_{\psi\in\mathcal{S}_n}\left\{\prod\limits_{i=1}^n\left(a_i+b_{\psi_i}\right)\right\}$$
provided that all eigenvalues of $A$, $B$ are real except possibly $\varepsilon=3$ eigenvalues of $B$. This improves on earlier results showing the same conclusion with $\varepsilon=1$ (Kovačec, 1999) and $\varepsilon=0$ (Fiedler, 1970).