It is a known fact that max range (and I know this is difference from max endurance, but not by much) is achieved independent of altitude, as per Carson. Because L/D ratio increases with altitude. There is no benefit by going high, especially when you also have to account for the fuel to get up there.
Therefore, and this is an educated guess, I'm pretty certain that a jet will have no endurance benefit by going to higher altitudes. Take off, level off at lowest obstacle clearance altitude and pull back to max endurance (which will be lower than Vbg).