The bounded-base tube-domain polydisc conjecture

Let $d\geq 1$, let $C\subset\operatorname{\mathbb{R}}^d$ be a bounded convex domain, and let $\Omega=C+i\operatorname{\mathbb{R}}^d$ be its tube domain. Bounded-base tube-domain conjecture. The domain $\Omega$ is biholomorphic to a bounded convex domain if and only if it is biholomorphic to the polydisc

$$\operatorname{\mathbb{D}}^d.$$

The conjecture concerns which tube domains with bounded convex bases admit bounded convex realizations; products of intervals give the model case, since their tube domains are biholomorphic to $\operatorname{\mathbb{D}}^d$. The source relates this question to a problem of Fornæss and K. T. Kim, and gives no resolution.