Existence of a totally real codimension-two linear slice of SO-star(5)

Let ${\textrm{SO}}^\star(5)$ denote the variety of orthogonal matrices with the relevant zero pattern, and let a totally real linear slice mean a codimension-two linear section whose points are all real. Existence conjecture. There exists a codimension-two linear slice of ${\textrm{SO}}^\star(5)$ which is totally real. The preceding discussion leaves constructing such a witness set for future research; no resolution is given here.