Is it true that there is no absolute morality?

[WisdomOfTheTrees]

To say that morality is not objective is a far cry from saying it is arbitrary.

Indeed. I've been thinking about this quite a bit lately.

I believe the problem is like shaking a gift box and knowing that there is something in the box but only being able to guess what it is (...or how it got there or who it's from, etc.) Morality may ultimately have an objective basis but that in no way necessitates that people can know everything about it. So I am willing to grant assumptions both atheistic and theistic approaches could share such as, the existence of conscience and the cultural contingency of all moral systems, etc. with the understanding that these are ultimately grounded in moral absolutes that transcend any individual or culture.

It is arbitrary. I explained that in the OP. A person's desires are arbitrary.