Thrust Limited?
RMC mentioned that he was maintaining M0.7 in a climb. Is it possible that the IAS dropped below the minimum drag speed? As the IAS continued decreasing with M0.7 being maintained in a climb, there is more drag (back side of the drag curve). So the aircraft becomes thrust limited and cannot maintain the climb, hence the A/P tripped. But RMC also mentioned that he was at least 6000' below Max Alt? I'm don't know much about the 737 FMC, so I'm just guessing here. Was the vertical speed decaying through the climb?