Complex illumination conjecture for the polydisc

Let $K\subseteq\mathbb{C}^n$ be a complex convex body, meaning the unit ball of a norm on $\mathbb{C}^n$ and, under the identification with $\mathbb{R}^{2n}$, invariant under multiplication by $e^{\theta i}$ for every $\theta\in\mathbb{R}$. Let $D$ be the closed unit disk in $\mathbb{C}$ and let $D^n=D\times\cdots\times D$ be the polydisc. The complex illumination number is denoted by $\operatorname{ill}(K)$.

Complex illumination conjecture. We have

$$\operatorname{ill}(K)\le 2^{n+1}-1.$$

Moreover, equality holds if and only if $K$ is a linear image of $D^n$.

The polydisc is the proposed complex analogue of the hypercube: the paper states that $\operatorname{ill}(D^n)=2^{n+1}-1$. The conjecture seeks the corresponding sharp universal bound and equality characterization for complex convex bodies; its resolution is not stated in the supplied text.