The characteristic-two Hermitian plus rank-one conjecture

Assume $\operatorname{char} K=2$, let $F$ be the fixed field of the involution on $K$ with $F\ne K$, and write $K=F[\omega]$ where $\omega^2+\omega+\beta=0$ for some $\beta\in F$ and $\overline{\omega}=\omega+1$. Let $A=B+\omega C$, where $B,C\in K^{n\times n}$ are Hermitian and $C$ has rank one. Characteristic-two Hermitian plus rank-one conjecture. The valuations of the principal minors of $A$ form a valuated $\Delta$-matroid. In characteristic two, the preceding skew-Hermitian plus rank-one statement is already true because the matrix is Hermitian; this is the additional case proposed by the source.