I had to look up at what temperature diamonds start to oxidize/burn[0]: Different sources say different things but apparently it's somewhere between 700°C and 900°C (depending on the exact conditions I suppose).
Even according to the article: "2,200 to 2,400 watts per meter per kelvin - roughly six times as conductive as copper.". It's way higher than copper in fact. Copper is ~400 W/(m·K)