The conjecture that I proposed in
is in fact known as the "double cap conjecture", as noted by Guillaume Aubrun. See for example
Here I propose a generalization. Suppose that the field of the unit vectors is not real, but complex. We can define a measure, which is induced by the measure of a real vector space. For example, the two-dimensional complex vectors $(a,b)$ are associatedto the 4-dimensional real vectors $(Re(a),Re(b),Im(a),Im(b))$. Thus, the measure on the latter space induces a measure on the former. What is the set of unit vectors with maximal area that does not contain pairs of orthogonal vectors. My conjecture: up to unitary rotations, the maximal set is the set of vectors $a$ such that $|a\cdot S|^2>1/2$, where $S$ is some unit vector. This is a generalization of the "double cap conjecture".
The question is: is there a easy proof of this conjecture from the "double cap conjecture"?