My experience with standard room award bookings at the Conrad Tokyo is, that they release availability later than other hotels (6-10 months in advance). And after the first release the same availability comes and goes. Maybe they release it step by step, maybe they release more, but take it back from time to time. So if you don't see availability now, it maybe was there before and maybe it was booked, maybe not.
Contacting them sometimes helps, sometimes not.
From my point of view it is the right strategy to book a Hilton as a backup (what you already did) and to check availability a couple of times every week. If you use the app for that, each check needs just a few seconds only.