I don't see anything on the BRS site stating to cure in freshwater, and all their videos use saltwater. Phosphates, if any, will come out regardless.
Many people have had large amounts of phosphates with Pakuni. I've come across threads with hobbyists soaking/curing in saltwater for months and still end up with high phosphates. These are other hobbyists experiences, not what BRS says. The amount of phosphates pakuni will vary from rock to rock since each rocks will vary in amount of dead stuff on it. Some people have are able to cure for 1-2 months and end up with little detectable phosphates, so not everyone will have the high phosphate issue.
SeaKlear has been used with Pakuni cure to shorten the amount of time to cure these rocks since nobody wants to wait several months for the regular cure. Dose daily when curing in saltwater and many people finish the cure in 3-4 weeks with zero detectable phosphates. Hobbyists claim that SeaKlear is reef safe, but I am uncertain.
Rocks leach phosphates in freshwater or saltwater. Phosphates will come out faster in water with 0 phosphates and then the rocks leaches slower as the phosphates come out to the water column, kind of like how osmosis works. Starting the curing process in freshwater helps loosen the dead stuff and you get the opportunity to rinse them off the dead stuff + remove them with water changes. The less dead stuff removed from the curing container with the pakuni, the less overall phosphates you will have to deal with. More water changes means faster rate of phosphates leaching from the rocks since the new water will have 0 phosphates. This is why I propose starting in freshwater for the first few weeks, then move to saltwater to finish the cure.
Starting the cure in saltwater helps start the cycle for beneficial bacteria that takes care of the nitrates. I'd rather take care of the phosphates first since there can be very high levels of them leach from pakuni, then beneficial bacteria later. This does not apply to BRS's reefsaver and probably not fiji.
Hope this makes sense. Its makes perfect sense to me unless there is some flawed logic that someone can point out.