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I believe someone recently tested this, and confirmed on here that they were able to start after midnight. It's not a phone-based system, so there's no need for them to wait until operators get to work. It's 100% online, so as soon as the clock says it's 60-days (based on Eastern Time), it supposedly works.
But unless you're going on a historically busy day (like a holiday), you don't need to be on the computer at 0-hour. They don't go as fast as ADRs used to, at 7am on the 180-day mark.