Boltyanskii's illumination conjecture for convex bodies

Let $K\in\mathcal{K}^d$ be a convex body, and say that a set of directions illuminates $K$ if every point of $\operatorname{bd}K$ is illuminated by at least one direction in the set; denote the smallest possible number of directions by $\operatorname{Ill}(K)$. Illumination conjecture. Every convex body $K\in\mathcal{K}^d$ can be illuminated by at most $2^d$ directions, and by fewer than $2^d$ directions if $K$ is not a linear image of a $d$-dimensional cube. This is a central problem in convex geometry concerning the illumination number of convex bodies. The source presents it as Boltyanskii's conjecture; no resolution is supplied in the text.