Suppose $A,A_1,\ldots,A_{n-2}$ (resp. $B$) are (resp. is) real positive-definite (resp. arbitrary) symmetric $n\times n$ matrices and denote by $D(\cdot,\ldots,\cdot)$ the mixed discriminant. We have the following well-known Aleksandrov-Fenchel inequality

\begin{equation}\label{e} D(A,B,A_1,\ldots,A_{n-2})^2\geq D(A,A,A_1,\ldots,A_{n-2})D(B,B,A_1,\ldots,A_{n-2}), \end{equation} with equality iff $A$ and $B$ are proportional.

Now my question comes: does this inequality and equality case still hold if we assume these matrices are complex Hermitian matrices rather than real symmetric ones? I guess this is the case but I am not able to find a reference. The standard textbook of Schneider only treats the real symmetric case.

Many thanks in advance!