We consider hyperbolic structures on the compression body C with genus-2 positive boundary and genus-1 negative boundary. Note that C deformation retracts to the union of the torus boundary and a single arc with its endpoints on the torus. We call this arc the core tunnel of C. We conjecture that in every geometrically finite structure on C, the core tunnel is isotopic to a geodesic. By considering Ford domains, we show that this conjecture holds for many geometrically finite structures. Additionally, we give an algorithm to compute the Ford domain of such a manifold, and a procedure that has been implemented to visualize many of these Ford domains. Our computer implementation gives further evidence for the conjecture.