The constant-width illumination conjecture in three dimensions

Let $\mathbf{K}$ be a three-dimensional convex body of constant width. Constant-width illumination conjecture. The illumination number satisfies $I(\mathbf{K})=4$. The known general upper bound is $I(\mathbf{K})\leq 6$, and the conjecture would imply a new proof of Borsuk's conjecture in dimension three; it remains open.