TLC5926 With No Load
While the Datasheet for the TLC5926 doesn't address this, and the internal topology isn't explicitly stated, the TI Engineer on the forum post gives what essentially is the same way to deal with the phenomenon:


  The TLC5940 is operating properly.   If you have the output programmed to sink current, but the output is open, the IC trys harder and harder to turn on the internal FETs which results in higher input current to the IC.  This will not damage the IC. If you want to reduce power consumption, program unused outputs to sink 0mA.


Of course, since the TLC5926 is a constant current on/off shift register, you can't program it to sink 0mA. You can how ever, simply turn off that individual(s) output if no load is connected. This would be the same way you would deal with a open-load detection.

Your system should check for the open or short circuit detection register, and disable the output as needed, in code.

As for knowing if the TLC5926 does have a higher no load current draw, you could, before connecting the LEDs (Or cutting the V+ to the leds), turn all outputs on, and measure the current draw, then connect the LEDs, and turn all the outputs on, and measure again.