The unbounded-face conjecture for Dirichlet domains in the bidisk

Let $\Gamma$ act on $H^{2}\times H^{2}$, and let $\gamma=(g_1,g_2)\in\Gamma$ with both $g_1$ and $g_2$ hyperbolic. A Dirichlet domain for the cyclic action generated by $\gamma$ is a domain bounded by equidistant hypersurfaces, called faces.

Unbounded-face conjecture. Every $\gamma=(g_1,g_2)$ where both $g_i$'s are hyperbolic admits a Dirichlet domain with more than two faces.

The preceding theorem shows that when the base point lies on the invariant flat of $\gamma$, the associated Dirichlet domain is two-faced. The claim concerns the existence of a Dirichlet domain with more than two faces, suggested by examples in which the base point does not lie on the invariant flat; the source does not establish whether this always occurs.