Nobody has solved it.

The most important thing to have when approaching this debate (or any other major philosophical debate) is epistemic humility. If you think that moral truths are obviously objective, or obviously subjective, consider the possibility that you're missing some of the complexity of the arguments on the other side.

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One thing worth doing here is laying out out the universe of major debates in meta-ethics, because a lot of times "subjectivism" gets conflated with a lot of different views.

Moral Realism is the view that there are some true moral facts. Say, murder is wrong.

Moral Anti-Realism is not just "subjectivism" but could be divided into some different views:

1. Non-cognitivists believe that moral statements are not the sorts of things that can be true or false. They may just be expressions of approval or emotion. So "murder is wrong", according to a non-cognitivist, might just mean something like "boo murder!". They key here is that moral statements are not beliefs. "murder is wrong" is neither true nor false on this view.

2. Error theorists believe that moral statements ARE beliefs that could be true or false, but that they are never true. So "murder is wrong", according to an error theorist, is false. It is false not because murder is good, but because the property "wrongness" doesn't exist in the world.

Subjectivism is technically a moral realist position by my definition. But Traditional moral realists are objectivists, believing that the truth or falsity of moral statements are mind-independent. So "murder is wrong" is either true or false no matter what I happen to think about murder. Non-objectivists believe moral truths are somehow mind-dependent. Subjectivism would say "murder is wrong" means something like "I disapprove of murder". But there are other non-objectivist positions. Cultural relativism, for example, would say "murder is wrong" means something like "my culture disapproves of murder".

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I'm not taking a view on any of these questions, I just wanted to lay out the landscape for you for further reading or maybe help you pinpoint your own views. Very important to remember that anybody who is giving glibly confident answers to these questions probably doesn't understand all the nuance because this stuff is not easy.

Arguments like Sam Harris’ have been refuted for decades in many ways. One is the is-ought gap - principally, you cannot logically derive what you ought to do (moral actions) from what is (biology, pleasure, happiness, etc).

Other notable refutations of objectivist morality like these are the Open Question Argument, and the Naturalistic Fallacy. So to answer your question, as with every debate in philosophy neither side has ‘solved it’, but the subjectivist side has never really had a problem entirely refuting arguments from the objectivist side

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.

I think Ayn Rand has solved morality.

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Morality is a phantasm, an illusory figment of the human imagination. For morality to be objective, there would need to be some universal independent standard to refer to, but no such thing exists.

Let's consider a car, which is travelling along a road. That car is going at a certain speed. As bystanders, we might guess at what speed the car is traveling, and we will have either a correct answer or an incorrect answer. The only reason that our answers can be correct or incorrect is because the car is traveling, and thus will be moving at a certain speed. It provides the standard by which our answers can be adjudicated. If there was no car, then to ask at what speed the car was traveling would not make sense.

In the case of morality, there is no car, which is to say, there is no standard. Taking your example here:

>Some say morality must be objective because we have biological needs like good health, which is universal, so anything that promotes good health in life should be objectively moral?

You haven't actually resolved the moral question. Why is it right to fulfill those biological needs? What makes that correct? By what standard, and by what authority is that standard applied to everyone? You can ask the same slate of questions of every purported moral system out there, and I, for one, have never seen convincing answers.