Carathéodory completeness conjecture for smooth bounded pseudoconvex domains

Let $D\subset\mathbb{C}^n$ be a bounded pseudoconvex domain with $C^\infty$ boundary. A domain $D$ is $c$-complete when its Carathéodory pseudodistance $c_D$ is a complete metric. Carathéodory completeness conjecture. Then $D$ is $c$-complete. This conjecture asks whether bounded pseudoconvex domains with smooth boundary are complete for the Carathéodory distance. The inverse implication from $c$-completeness to $c$-finite compactness is stated as an open question for domains in $\mathbb{C}^n$, and this conjecture has been open for more than 30 years.