The Gromov-hyperbolic tube-domain ball conjecture

Let $d\geq 1$, let $C\subset\operatorname{\mathbb{R}}^d$ be a convex domain, and let

$$\Omega=C+i\operatorname{\mathbb{R}}^d$$

be its tube domain. Assume that $(\Omega,K_\Omega)$ is Gromov hyperbolic; equivalently, $C$ is unbounded and $(C,H_C)$ is Gromov hyperbolic. Gromov-hyperbolic tube-domain conjecture. The domain $\Omega$ is biholomorphic to a bounded convex domain if and only if it is biholomorphic to the unit ball.

For unbounded bases, the source describes the situation as more mysterious and gives a non-homogeneous example of a tube domain biholomorphic to a bounded convex domain whose Kobayashi metric is not Gromov hyperbolic. Thus the conjecture isolates the Gromov-hyperbolic case, but the source provides no resolution.