The Aleksandrov-Fenchel inequality of mixed discriminants for Hermitian matrices 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!
 A: The Van der Waerden Conjecture for Mixed Discriminants, by Leonid Gurvits (2004), proves a generalized Alexandrov-Fenchel inequality for semidefinite $n\times n$ Hermitian matrices: 
Theorem 5.2, with equation (21), that for the special case $\alpha=(1,1,\ldots,1)$, $\alpha^{(1)}=(2,0,1,\ldots 1)$, $\alpha^{(2)}=(0,2,1,\ldots 1)$ reduces to the desired inequality with an extra factor $\exp(-2n^n/n!)$ on the right-hand side. 
Not good enough, obviously, but I have not found a proof for the Hermitian case without this reduction factor.
A: The answer is yes, as shown by Ping Li, theorem 3.1.
https://arxiv.org/pdf/1710.00520.pdf
or by Shenfeld and Van Handel
https://arxiv.org/pdf/1811.08710.pdf
Moreover in the original paper [A. D. Alexandroff. “Zur theorie der gemischten volumina von konvexen Körpern. IV. Die gemischten Diskriminanten und die gemischten Volumina.” In: Rec. Math. [Mat. Sbornik] N.S. 3(45).2 (1938), pp. 227–251], second page lines 20-23, Alexandroff claims that his proof for the real case works for the Hermitian one too.
