The definition of integrity requires honesty. Do you think it is honest to tell people "there are 38 rooms left" when you know 38 is a random number?
The definition also requires you to act morally. Do you think that anything legal is moral, and vice versa?
Slavery was legal. German extermination camps were legal. Were these moral?
Smoking weed is illegal. Is it immoral?
Morality is subjective but not in the manner you're describing. It's not arbitrary. Our values are conjured out of thin air but for reasons that DO involve the betterment of humanity.
Slavery isn't subjective, and extermination camps sure as hell are not either.
How is slavery or extermination camps not subjective? The slave owner may think slavery is good thing but the slaves may think its horrible.