Four minutes from 35,000 feet is over 8,000 feet per minute. If the report was triggered by the same event that caused the aircraft loss, that's a rate of descent faster than would be achieved by the pilots in an intact aircraft in an emergency descent.
It would also be no surprise that no radio calls were received, as the pilots wouldn't be wearing radio headsets in HF airspace (you only put them on if you are called over SELCAL, or intend to transmit a position report.)