What is the object experiences the greater magnitude gain or loss of heat during the time the system takes to reach thermal equilibrium?

The correct answer to this question is C.

In thermal equilibrium, it is observed that a higher temperature object which is in contact with a lower temperature object will transfer heat to the lower temperature object. The objects will approach the same temperature, and in the absence of loss to other objects, they will then maintain a constant temperature.

They are then said to be in thermal equilibrium. Thermal equilibrium is the subject of the Zeroth Law of Thermodynamics. This implies that there is a measurable property that can be considered to be the same for A and B, a property upon which heat transfer depends.