Sorry - wrong.
Well sort of-not completely 
What they taught us at Pensacola many decades ago. Bolding mine.
The effect of altitude is to reduce the engine airflow and power output and supercharging is necessary to maintain high power output at high altitude. Since the basic engine is able to process air only by the basic volume displacement, the function of the supercharger is to compress the inlet air and provide a greater weight of air for the engine to process. Of course, shaft power is necessary to operate the engine driven supercharger and a temperature rise occurs through the supercharger compression. The effect of various forms of supercharging on altitude performance is illustrated in figure 2.17.
The unsupercharged-or naturally aspirated-engine has no means of providing a manifold pressure any greater than the induction system inlet pressure. As altitude is increased with full throttle and a governed RPM, the airflow through the engine is reduced and BHP decreases. The first forms of supercharging were of relatively low pressure ratio and the added airflow and power could be handled at full throttle within detonation limits. Such a “ground boosted" engine would achieve higher output power at all altitudes but an increase in altitude would produce a decrease in manifold pressure, airflow, and power output.
More advanced forms of supercharging with higher pressure ratios can produce very large engine airflow. In fact, the typical case of altitude supercharging will produce such high airflow at low altitude operation that full throttle operation cannot be utilized within detonation limits. Figure 2.17 illustrates this case for a typical two-speed engine driven altitude supercharging installation. At sea level, the limiting manifold pressure produces a certain amount of BHP. Full throttle operation could produce a higher MAP and BHP if detonation were not the problem. In this case full throttle operation is unavailable because of detonation limits. As altitude is increased with the supercharger or “blower” at low speed, the constant MAP is maintained by opening the throttle and the BHP increases above the sea level value because of the reduced exhaust back pressure. Opening the throttle allows the supercharger inlet to receive the same inlet pressure and produce the same MAP.
Finally, the increase of altitude will require full throttle to produce the constant MAP with low blower and this point is termed the “critical altitude” or “full throttle height.” If altitude is increased beyond the critical altitude, the engine MAP, airflow, and BHP decrease.
The critical altitude with a particular supercharger installation is specific to a given combination of MAP and RPM. Obviously, a lower MAP could be maintained to some higher altitude or a lower engine speed would produce less supercharging and a given MAP would require a greater throttle opening. Generally, the most important critical altitudes will be specified for maximum, rated, and maximum cruise power conditions.
A change of the blower to a high speed will provide greater supercharging but will require more shaft power and incur a greater temperature rise. Thus, the high blower speed can produce an increase in altitude performance within the detonation limitations.
The variation of BHP with altitude for the blower at high speed shows an increase in critical altitude and greater BHP than is obtainable in low blower. Operation below the high blower critical altitude requires some limiting manifold pressure to remain within detonation limits. It is apparent that the shift to high blower is not required just past low blower critical altitude but at the point where the transition from low blower, full throttle to high blower, limit MAP will produce greater BHP. Of course, if the blower speed is increased without reducing the throttle opening, an “overboost” can occur.
Since the exhaust gases have considerable energy, exhaust turbines provide a source of supercharger power. The turbosupercharger (TBS) allows control of the supercharger speed and output to very high altitudes with a variable discharge exhaust turbine (VDT). The turbosupercharger is capable of providing the engine airflow with increasing altitude by increasing turbine and supercharger speed.
Critical altitude for the turbosupercharger is usually defined by the altitude which produces the limiting exhaust turbine speed.
The minimum specific fuel consumption of the supercharged engine is not greatly affected by altitudes less than the critical altitude. At the maximum cruise power condition, specific fuel consumption will decrease slightly with an increase in altitude up to the critical altitude. Above critical altitude, maximum cruise power cannot he maintained but the specific fuel consumption is not adversely affected as long as auto lean or manual lean power can be used at the cruise power setting.
Is speed dependent as well due ram recovery at the intake. Consider combat aircraft carrying underwing stores ie drop, tanks, bombs etc