Orthogonality conjecture for low-multiplicity spherical zone coverings

Let $d\geq 3$ and $n\geq 1$ be integers, and let $S^{d-1}$ be covered by the union of $n$ congruent spherical zones. Suppose each point of $S^2$ belongs to the interior of at most two zones. Orthogonality conjecture. Then $n\leq d-1$. Moreover, if $n=d$, then the $d$ congruent zones are pairwise orthogonal.

This conjecture concerns coverings whose pointwise interior multiplicity is at most two and is motivated by the Euler–Poincaré analysis of arrangements of central great spheres. The source does not provide evidence that either assertion has been resolved.