Extremal great-circle distribution conjecture for six, nine, and fifteen circles

Let $r_{\rm gc}(n,\mathbb{S}^2)$ be the largest radius such that every cell in a tiling of the unit sphere $\mathbb{S}^2$ induced by $n$ great circles contains a spherical cap of that radius. For $n=6,9,15$, let $\rho_n$ denote the inradius of the corresponding congruent Coxeter spherical triangle.

Extremal great-circle distribution conjecture.

$$r_{\rm gc}(6,\mathbb{S}^2)=\rho_6,\qquad r_{\rm gc}(9,\mathbb{S}^2)=\rho_9,\qquad r_{\rm gc}(15,\mathbb{S}^2)=\rho_{15}.$$

The displayed Coxeter arrangements establish the corresponding lower bounds; the conjecture asserts their optimality. The source does not state a resolution, so the conjecture remains open.