Kalai–Wilson conjecture on the largest orthogonality-free subset of the sphere

Let $A\subseteq S^2$ be a measurable subset containing no two orthogonal directions. Its measure is normalized by $|S^2|=1$. Kalai–Wilson's conjecture. The maximal measure of such a subset is attained by two open caps of geodesic radius $\pi/4$ around the south and north poles. This problem was posed by Witsenhausen in 1974; the conjecture had not been solved or disproved in the source context.