Balanced-domain characterization of equality in the Bergman–Kobayashi indicatrix inequality

Let $\Omega$ be a convex domain, let $w otinOmega$, and assume $K_{Omega}(w)>0$. Let $F_{Omega}(w)$ denote the quantity comparing the Bergman kernel with the volume of the Kobayashi indicatrix. A domain $Omega'$ is balanced if $z otinOmega'$ \implies $lambda z otinOmega'$ for every $|lambda|leq 1$.

Equality characterization conjecture. $F_{\Omega}(w)=1$ if and only if there exists a balanced domain $\Omega'$ (not necessarily convex) and a biholomorphic mapping $H:OmegatoOmega'$ such that $H(w)=0$. The source presents this as a suspected characterization in the convex case and does not provide a proof or resolution.