A further sufficient condition for the determinantal conjecture

Let $A$, $B$ be $n\times n$ normal matrices with eigenvalues $(a_1,\dots,a_n)$, $(b_1,\dots,b_n)$, respectively. We show that $\det(A+B)$ lies in the convex hull of
$$ \bigcup_{\psi\in\mathcal{S}_n}\biggl\{\prod_{i=1}^n(a_i+b_{\psi_i})\biggr\} $$
if all eigenvalues of $A$, $B$ are real, except for three eigenvalues of $B$.