At least for me, it seems that the assumption that there is no absolute source of morality besides religion is correct. Human morality has changed a lot throughout history, and lots of people have tried to dictate morality across borders and across time. The only ones who have succeeded in the slightest are religious leaders.

The argument is generally that one specific religion provides a source of absolute morality. The existence of conflicting religions does not invalidate that. It provides one source of absolute morality, not necessarily the only source of absolute morality. Anyone can claim something is a source of absolute morality. I can claim a magic 8-ball is a source of absolute morality. It does not mean that people will accept it, but I can claim it.