Comparing the Carath\'eodory Pseudo-Distance and the K\"ahler-Einstein Distance on Complete Reinhardt Domains

We show that on a certain class of bounded, complete Reinhardt domains in ${\mathbb{C}}^n$ that enjoy a lot of symmetries, the Carath'eodory pseudo-distance and the geodesic distance of the complete K"ahler-Einstein metric with Ricci curvature $-1$ are different.