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.