arXiv:2610.36124v1 Announce Type: new Abstract: We study the explicit bounded pseudoconvex domain \[ \Omega=\{(z_1,z_2,w)\in\mathbb C^3: |w|^2+|z_1z_2(z_1-z_2)|^2+|z_1|^8+|z_2|^8<1\}. \] Its boundary is real analytic and has finite D'Angelo type everywhere. We prove two assertions about the invariant metrics of this domain. First, $(\Omega,k_\Omega)$ is not Gromov hyperbolic; in fact arbitrarily fat Kobayashi geodesic triangles occur in every neighborhood of the type-$8$ point $(0,0,1)$. Second, the Carath\'eodory--Reiffen and Kobayashi--Royden metrics are not uniformly comparable. Along an explicit family of tangential directions the Carath\'eodory metric tends to zero while the Kobayashi metric has a positive limit. For the corresponding distances there are points $x_j,y_j\to(0,0,1)$ such that $c_\Omega(x_j,y_j)\to0$ while $k_\Omega(x_j,y_j)\to\infty$. This gives, among bounded real-analytic pseudoconvex domains, an affirmative answer to Problem~7.3.2 of the Several Complex Variables Problem List.
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