Constant Current IC Driving a String of ICs
A Constant Current LED Driver like the TI TLC5916 can be used like that. It depends on VLed, the LED Voltage, to be roughly 0.5~2 Volts above the Vf of the LEDs. It will then use it's N-Channel Fet or Transistor and feedback circuit to limit the output to the set current. The current would be the same regardless of how many leds are in Series.


  VO must not be too high as this causes excess power dissipation inside the current sink. However, VO also must not be too low as this does not allow the full LED current (Figure 4). With VLED = 12 V:
  VO = 12V–(3×3.5V)= 1.5V


A big difference like you want, 7 Leds on one, 1 Led on another, would be bad. 7x 2.0 Vf led + 1 Volt control voltage = 15 Volts minimum. Dissipating 1 Volt @ 20mA = 0.02 Watts is fine. If that same voltage is used for 1 led, the driver would have to deal with 13 volts @ 20mA = 0.26 Watts to dissipate. That would push the TLC5916 close enough to it's heat limit of 150°C for that single channel itself (See data sheet for numbers).

You could do two things to compensate. One, use different voltages for VLed. That's unnecessarily difficult. Two, use resistors to help dissipate the difference. That defeats the purpose.

So no, it wouldn't help.