Let $A$ be an $n \times n$ hermitian matrix with $n \geq 3$. I am trying to prove the following inequality involving its minors $$\left| \sum_{k=3}^n A_{3k} A_{[12k],[123]} \right| \leq \sqrt{\sum_{i < j} |A_{[13],[ij]}|^2}\sqrt{\sum_{i < j} |A_{[23],[ij]}|^2},$$ where the square brackets in subscripts contain the numbers of rows and columns involved in the given minor. The inequality holds trivially (and saturates) for diagonal matrices and passes all numerical tests with flying colours, but I am unable to crack it in the general case. It looks Cauchy-Schwarz-esque, but that resemblance did not lead me anywhere.
Let me add that the above inequality can be written equivalently as $$|\langle u_1 \otimes u_2 \otimes A u_3, A u_1 \wedge A u_2 \wedge A u_3 \rangle | \leq \tfrac{1}{2} \|A u_1 \wedge A u_3 \| \|A u_2 \wedge A u_3 \|$$ for any hermitian $A \in B(\mathbb{C}^n)$ and any pairwise orthonormal $u_1,u_2,u_3 \in \mathbb{C}^n$, where the wedge products are not normalized, i.e. $u \wedge v = u \otimes v - v \otimes u$ and similarly for the triple wedge product. The inner product on $(\mathbb{C}^n)^{\otimes 3}$ is of course the natural one.
Any hint would be much appreciated!
