Does changing control of an object update that object's timestamp?

Question

This is a follow-up question to does gaining control of an aura gaining control of a permanent give me the permanent?

1. Player A has a creature.
2. Player B casts Control Magic on it. B controls the creature.
3. Player C also casts Control Magic on it. C controls the creature due to the enchantments' timestamps.
4. Player D casts Steal Enchantment on player B's Control Magic.

Does player D or C now control the creature now? Player C's Control Magic is more recent, but does B's Control Magic timestamp get updated because of Steal Enchantment?

Answer 1

Player C controls it

613.6a A continuous effect generated by a static ability has the same timestamp as the object the static ability is on, or the timestamp of the effect that created the ability, whichever is later.

613.6c An object receives a timestamp at the time it enters a zone.

Even though you gain control of it it stays in the same zone, the battlefield, but it is an aura so there are more rules to it

613.6d An Aura, Equipment, or Fortification receives a new timestamp at the time it becomes attached to an object or player.

But it is still enchanting the same creature so it does not receive a new timestamp meaning player c still controls it