Ah, I think I have just figured out the problem.
The CTRL input is essentially a 1.5k resistor to ground. You need to be able to get at least 4.2V on it in order to turn the LED off.
Assuming a 5v supply to the TLC5940, that means you need <0.8v across the pullup resistor. This means you need the pullup resistor to be no more than 285R (That is 1.5k * 0.8/4.2).
To be on the safe side, pick a pullup resistor of say 220 ohms, and set the IREF resistor for the TLC5940 to:
V = IR
I = 5/220 = 23mA
Rref = 31.5*1.24/Iout = 39/23mA ~= 1.8k
So
Rref < 1.8k
Use a pullup of 220ohm, and Reference resistor of 1k
(P.S. You are corrent, the Ref pin doesn't need to be connected, it is essentially a 5v supply)