Most (all?) large-cabin long-range business jets are equipped with Emergency/Automatic Descent Mode (EDM or ADM, depending on the manufacturers nomenclature). Above a certain altitude, say 30,000’, the system is armed to recognize a sudden drop in cabin pressure. The autothrottles reduce power to idle, the autopilot turns the plane 90-degree to the left and descends to (usually) 15,000’. It will level off there and increase thrust to maintain 250 KIAS until (hopefully) a pilot is able to take back control.
Unfortunately, this airplane type has neither autothrottles nor EDM/ADM. Because of this, pilot intervention would be necessary for the plane to begin a controlled descent from 34,000’. Once captured/level, selecting a lower altitude and commanding a vertical mode are required to descend, even in airplanes equipped with autothrottles.
Pretty safe to say that pilot incapacitation (hypoxia or medical) is to blame in this tragic accident.