We did a behind the scenes ship tour on the Equinox in June. After seeing how everything is laundered there is no way anything can be washed in cold water. All laundry from passengers are washed together in huge commercial machines. They do use detergents that are hypoallergenic with little to no scents. Think about how the laundry department washes and dries thousands of sheets, towels and linens per day, how could they take any passenger's bag of laundry and honor the cold water only requests.
We are Elite so we get the one bag of free laundry, but my wife will not send out and of her delicates. She does a lot of hand washing in the cabin and uses something she buys from Amazon that are sheet detergent.