The skew-Hermitian plus rank-one Hermitian principal-minor conjecture

Let $K$ be a valued field with an automorphic involution preserving the valuation. Let $A=B+C$, where $B\in K^{n\times n}$ is skew-Hermitian and $C\in K^{n\times n}$ is Hermitian of rank one. The principal-minor valuations of $A$ are the function assigning to each $S\subseteq[n]$ the valuation of the principal minor indexed by $S$. Skew-Hermitian plus rank-one Hermitian conjecture. The valuations of the principal minors of $A$ form a valuated $\Delta$-matroid. This would extend the corresponding result for symmetric and skew-symmetric matrices and the proved skew-symmetric plus rank-one symmetric case to valued fields with involution; it is stated as open in the source.