No solution, only more or less convincing arguments.

I do not think the question is as clear cut though as objective vs subjective. It is possible also to argue that morality is a social property. To see morality as a matter of the individual subject would then be wrong. However, neither would the morality be founded in a universal nature. The social laws and conventions are then imitated, adopted, reproduced through the individual human. In a sense that is an objective morality, not a choice or something individual, but neither is it universal.

If we accept this one can debate depth. For example, if our moral intuitions about who or what to blame for an unprovoked murder, or different moral status of children, can be traced back to some conventions from millennia ago, what does that imply for the present? Can we elect to switched the moral system that plays out within us or not? There is a bootstrap problem here, which I know some philosophers like Agnes Callard are thinking of. Questions about the truth in traditions are also found here, truths that are not simply matters of scientific scrutiny.

I find that many deeper debates about morality end up in questions about purpose or telos. Is everything arbitrary, or has the human creature been imbued with some purpose. Even the Sam Harris approach to look at biology and survival and reproduction ends up there, attributing meaning to suffering. It makes the God question also inevitable. Alasdair MacIntyre has looked into telos and why some given feature of the universe we live in is likely to have granted us with an objective purpose.

Lots more can be said, my point merely is that once we look deeper into the question you pose, another set of issues are encountered, which challenges the question and what morality is or can be.