Continuous extension conjecture for complex geodesics of strictly convex domains

Let $\Omega$ be a bounded convex domain with $\mathcal{C}^1$-smooth boundary that is $\mathbb{C}$-strictly convex. A complex geodesic of $\Omega$ is an isometric holomorphic embedding of the unit disc $\mathbb{D}$ into $\Omega$ for the Poincaré and Kobayashi distances.

Continuous extension conjecture. Every complex geodesic of $\Omega$ extends continuously to $\overline{\mathbb{D}}$.

This conjecture proposes a natural boundary-regularity condition ensuring continuous extension of complex geodesics, without requiring finite-type boundary points or Gromov hyperbolicity. The source presents it as a direction suggested by known extension results; its resolution is not indicated here.