The usual simple statement of geodesics is that when moving from point $(x_o,y_o,z_o,t_o)$ to point $(x_1,y_1,z_1,t_1)$ the object will follow the path of greatest proper time experience, i.e., a clock traveling the geodesic will record more seconds than one going another path. I believe that is true even if you allow rocket thrusters or other energy addition devices.
But what if you consider a circular orbit which, after 1 year, results in going from $(0,0,0,0)$ to $(0,0,0,1)$? Wouldn't the proper time be maximized by simply sitting still, i.e., traveling only through time and not space? The orbiting clock would presumably run slower just as the orbiting clock in the international space station runs slower than one on Earth. In fact, if you want a physical model, assume a clock on a 400 km tall flag pole synchronized with the ISS during a pass-by (the ISS orbits at 400 km above the planet surface). One orbit later, wouldn't the ISS clock record less time than the flag pole clock?
Is that a violation of the concept "orbits are geodesics, for which the shortest distance between two 4D points is the path of greatest time"?