Suppose you control a Soul Tithe, enchanted on your own Conjured Currency.

At the beginning of your upkeep, if you choose to exchange control of Conjured Currency and a permanent your opponent controls, what will happen when Soul Tithe's ability resolves?

Obviously you cannot sacrifice a permanent you do not control, but my confusion arises from what exactly “that player” refers to in the cardtext. Does it refer to the player whose upkeep Soul Tithe's ability triggered on, or the current controller of the enchantment? Depending on which it is, this will change the outcome of the triggered ability.

Does “that player” refer to the current controller, or the player whose upkeep the ability triggered on?