Illumination conjecture for 1-unconditionally symmetric cap bodies

Let $\mathbf{K}\subset\mathbb{E}^d$ be a $1$-unconditionally symmetric cap body, meaning that $\mathbf{K}$ is symmetric about each coordinate hyperplane of $\mathbb{E}^d$, and let $d\geq 5$. Illumination conjecture for 1-unconditionally symmetric cap bodies. Every such cap body can be illuminated by $2d$ directions.

The preceding theorem gives the weaker bound $I(\mathbf{K})\leq 4d$, while the conjectured $2d$ estimate would establish the general illumination bound for this class and is proposed as a sharper result.