No, the only thing moving faster is the air the aeroplane is flying through. It must move faster in order to remain stationary reference the ground and meet the "nil wind" requirement. As long as the nil wind requirement is met, the aeroplane can fly with the same TAS as a lower aeroplane and cover the same ground distance.
Basically the previous posters are correct that something that is higher has more distance to travel because of the increasing circumference around the Earth as you get further from the centre of the Earth. What they missed was that this extra distance is entirely offset by the extra speed* the air has in order to remain stationary over the ground. It then comes back to the basic formula of TAS + wind = ground speed and ground speed / time = distance. With nil wind the TAS = ground speed and both aircraft travel the same distance over the ground.
Of course in real life the wind can't remain completely stagnant like that as you go higher. So the question is purely academic, and I think it's just designed to see whether you understand what TAS is. You're probably not meant to (and don't need to) start considering extra distance travelled the further you get from the centre of the Earth.
*As measured by an outside observer. The speed reference the ground is still zero.