Yes, the current can be adjusted to zero. For example, with the 1k resistor on the output, I read 0.0mA on my multimeter, and the led lit when I rotate the current pot down to zero.
LE: and I also read 0.0mA without the 1k resistor.
LE2: ... the fine tuning of the CC and CV loop capacitors is necessary or the capacitors does not allow any tuning ?