Can an LED achieve full brightness in 40 µs?
(1) LED on times for phosphor LEDs are in the 100'2 of nanoseond range

(2) Turn on times for non-phosphor LEDs are typically in 10's of nanosecond range if driven correctly.

Average current = Peak_Current x time_on / ( time_on + time_off )
 Peak current is assumed to be "steady".

(3) Brightness when mutiplexed 

     = B_DC x time_on / ( time_on + time_off ) x k    


Where B_DC is the brightness when the LED is operated at this PEAK current when DC is used and k = a factor relating to loss of efficiency with current, change of efficincy with die temperature etc.
 Initially k=1 is close enough. 

or Brightness using average current  = 

     = B_100% x k     at average current


(4) Modern phosphor LEDs have an allowable peak current 20% to 100% higher than the rated DC current.
 ie you cannot usefully multiplex modern phosphor LEDs directly.

(5) SOME modern LEDs MAY allow higher peak/rated current ratios but
 you should check the data sheet in EVERY case.

(6) There is a way to multiplex LEDs to allow high peak multiplex currents when the actual LEDS have low allowable peak/rated current ratios.
 It takes more circuitry and/or design effort. Few people do this AFAIK

There are various possible implementations but the basic method is to multiplex power (LED drive) to an energy store and then drive the LED from the energy store in such a way that LED current is about constant.

An "energy store" can be a capacitor or an inductor, plus supporting circuitry.

(a) Multiplex into capacitor across LED directly. Input desired average current. LED will stabilise at appropriate voltage for the average current. Energy is lost in the driver due to unavoidable I^R loss.
 Capacitor must be large enough to prevent LED current rising above rated value during recharge pulses.
  The capacitor increases the turn off time to at least a few multiplex cycle periods and probably 5 to 10 multiplex cycle periods, and maybe much longer at very high multiplex ratios.
 Turn on time is under the control of the designed but will also usually be slowed to several mutiplex cycle times.

(b) Multiplex into eg inductor in series with LED to ground. Reverse diode from input to ground. This is effectively a buck converter.