I wouldn't say either is particularly more related to diligence than the other. Other than being considered positive adjectives, they just mean different things. If I had to choose, I would say integrity - but I wouldn't consider them synonyms.

A criminal can be diligent without any integrity.

As both previous posters noted neither is directly related, but I could make a case that "initiative" is tangentially related. People who are diligent generally also show "initiative."

I agree with Tim - diligence is more about thoroughness and care. Initiative is about taking a first step forward or taking the decision to go forward with something. And integrity is a value relating to honesty.