Berndt's generalized Hlawka inequality for positive definite matrices

For $n\geqslant 3$, let $A_1,\ldots,A_n$ be positive definite. For each $k=1,\dots,n$, define the elementary symmetric determinantal polynomial

$$s_k:= \sum_{1 \leqslant i_1 < i_2 < \cdots < i_k \leqslant n} \det(A_{i_1} + \cdots + A_{i_k}).$$

Berndt's conjecture. The following generalization of the Hlawka inequality holds:

$$s_n + s_{n-2} + \cdots \geqslant s_{n-1} + s_{n-3} + \cdots.$$

This conjecture generalizes the classical Hlawka inequality and was the original motivation for the paper. The supplied source does not indicate whether it has been resolved.