The separation conjecture for convex bodies

Let $\mathbf{K}$ be a convex body in $\mathbb{E}^d$, $d\geq 3$, and let $\mathbf{o}$ be an interior point. A face is the intersection of $\mathbf{K}$ with a supporting hyperplane. Separation conjecture. There exist $2^d$ hyperplanes in $\mathbb{E}^d$ such that every face of $\mathbf{K}$ can be strictly separated from $\mathbf{o}$ by at least one of them; moreover, $2^d$ hyperplanes are needed only when $\mathbf{K}$ is the convex hull of $d$ linearly independent line segments intersecting at the common relative interior point $\mathbf{o}$. This is an equivalent formulation of the illumination conjecture and remains open in general.