Existence of real witness sets for special orthogonal groups

Let $n$ be a positive integer, and let a real witness set for an algebraic variety mean a witness set whose points are real. Existence conjecture. For any $n$, the algebraic group $\operatorname{SO}(n)$ admits some real witness set. The authors report computational evidence for $\operatorname{SO}(4)$ and $\operatorname{SO}(5)$, while noting that the compactness of $\operatorname{SO}(n)$ over the real numbers ensures that witness sets with no real solutions also occur. The conjecture asserts existence, not a uniform bound on the number of real solutions.