It is argued that the early geomagnetic field must have been maintained in a relatively thin molten silicate ocean situated just above Earth's liquid metal core. Here, we carry out detailed numerical models of such a thin layer dynamo. When coupled to models of Earth's thermal evolution and geophysical turbulence, we find the electrical conductivity of the basal magma ocean (BMO) fluid is too low for dynamo action to have occurred, likely requiring alternative mechanisms for generating Earth's ancient magnetic field.