The determinant of the sum of normal matrices

Given two normal matrices $A,Bin M_n({mathbb C})$
whose respective spectra are $(alpha_{1},ldots,alpha_{n})$ and
$(beta_{1},ldots,beta_{n})$, is it true that $det(A+B)$ belongs to
the convex hull of the set of numbers
$$prod_{i=1}^n(alpha_i+beta_{sigma(i)}),$$
as $sigma$ runs over the set ${mathfrak S}_n$ of permutations of ${1,ldots,n}$ ?

Nota. It is known (see Exercise 101) that the trace of $AB$ belongs to the convex hull of the
set of numbers
$$sum_{j=1}^nalpha_{j}beta_{sigma(j)},qquadsigmain {mathfrak S}_n.$$