This is the kind of value a population can collectively hold until they look around and see the culture doesn't value it anymore. Moral integrity stopped being a cultural value that mattered here before I was even born, if it ever really did matter for anyone except the "common" man.
Honestly, I don't care about what the culture does. I act with integrity because of my values and who I want to be, not because I'm under any illusions about how many of my peers will do the same. It is, in my opinion, the only way to live well.
You should care what your culture does.