Icosahedral optimality conjecture for six unit vectors in three-dimensional real space

Let $v_1,\dots,v_6$ be unit vectors in $\mathbf{R}^3$, and suppose one of the vectors lies at the north pole. An optimal configuration maximizes the minimum absolute determinant among the six vectors. Icosahedral optimality conjecture. An optimal configuration is given by the six vertices of an icosahedron contained in the northern hemisphere. This conjecture proposes a special-polyhedron configuration for the maximin determinants problem beyond the case of $n+1$ vectors in $\mathbf{R}^n$; the surrounding discussion explains that the corresponding problem for more than $n+1$ vectors remains unresolved.