Open chromatic number of the orthogonality sphere

Determine or estimate the least number of open sets containing no orthogonal pair that cover the sphere $S^{n-1}$, with particular emphasis on the exact value for $S^2$.

Background

For literal orthogonality on $S^{n-1}\subseteq\mathbb{R}^n$, cliques are orthonormal systems, so the clique number is known to be $n$. The corresponding open chromatic number requires a cover by open orthogonality-free sets and is a stronger covering invariant.

For $S^2$, the paper gives the bounds $4\leq\chi(S^2,\perp)\leq(S^2,\perp)\leq24$, using the known unrestricted chromatic number and an explicit open cover construction. The exact open chromatic number, and its higher-dimensional behavior, remain unresolved.