LiveATC archive has a couple of snippets from the pilots. Sounds like TCAS working on both at the time - 581 reported RA, 576 reported clear of conflict returning to FL390.
I was working 576 a couple of hours later on descent into PH and had been told TCAS US by that time... Perhaps they're a one use item?
Originally Posted by Andu
someone can (will) correct me if I'm wrong, but I understand that this particular incident was caused by one aircraft being cleared to climb to 2000' below the opposite direction traffic's cruising altitude
ATSB bulletin (supported by the snippets available on LiveATC) says that 576 was level at FL390, 581 at FL380 requested FL400 which was immediately issued, then cancelled a few seconds later. 581 responded to RA whilst on descent back to FL380. Climb re-issued some minutes later.