Starlikeness conjecture for Cassinian balls in starlike domains

Let $G\subset\mathbb{R}^n$ be a domain that is starlike with respect to $x\in G$, and let $B_{c_G}(x,r)$ denote the Cassinian ball of radius $r$. Starlikeness conjecture. For every $r>0$, the ball $B_{c_G}(x,r)$ is starlike with respect to $x$. This is a proposed extension of convexity results to starlike domains, and the paper does not establish it.