The possibility to believe in free will != free will.
Most modern science I've seen seems to come down on the side that free will is an illusion our brain plays on us to increase our sense of autonomy. We fool ourselves into mixing up the lines of causality to insert ourselves into the chain of events more than we actually were.
If it is possible for you to imagine free will, its because your subconscious is deciding to surface those thoughts to your conscious and you rationalize them from there based on what you think you know. However in the end, your brain is just a wet circuit running through some code. By the time you see the result on the screen, the computer's already processed it for you.
When modern scientists stand on the shoulders of giants, they sometimes risk undermining philosophical foundations on which the value of their work depends but which they do not understand.
Consider these four possibilities:
The first two scenarios are irrelevant, since there is no potential for us to influence them in any way. But in either of the latter 2, we are free to move between correct belief and incorrect rejection. So if belief is possible, why would anyone choose to be wrong?
Our ability to influence the first two in no way discounts them as plausible options. Reality doesn't ask our permission to be the way it is. Also, I disagree that the first two are necessarily mutually exclusive and require being broken up. They simply become individualized to the person's cognitive abilities, processes and their experiences which lead them to tend toward one side of the argument or the other.
Lets for a second consider the simulation argument. If we're simply simulated beings inside one of many simulated universes, its entirely reasonable that we have no free will, but because the mechanics of nature are emergent the possibility exists that some agents will become causally determined to believe we have free will, while others are causally determined to believe we don't, both based on their different experiences within the simulation where they absolutely do not have free will. Their opinions or ability to influence that outcome make no difference. In fact, with cognitive frameworks being a continuum of experience, there's nothing saying an individual won't change their mind on the subject many times, but make no mistake thinking that means they caused that change to happen, that change was simply caused through them by an infinite chain of events leading up to it.
For free will to exist, you'd need to employ some form of dualism which doesn't seem to hold up to scientific scrutiny or fit with the things we do know.
What I am arguing is not that free will itself can be proven, but that belief in it provably maximizes the likelihood of being correct. Under pure determinism, we contribute no variability, which means that for any given moment the minimum likelihood of us being correct in our belief is equal to the maximum likelihood. Under free will, the minimum possible likelihood is 0 if we choose to reject free will, and the maximum likelihood is 1 if we choose to believe in it.
I offer the hypothesis that both free will and time are illusions of consciousness. Consciousness could be a state of matter or it could be something quantum but at this point in time we don't fully understand it. What we know is energy (and information) cannot be created or destroyed. This means the universe itself while it is expanding, it's not going to have any more energy in it in 10 billion years than it had in the first seconds of the big bang.
So in order to answer whether or not free will exists you have to answer whether time itself exists. If time is an illusion which only exists because conscious entities exist in a particular universe, then free will might be part of the illusion. The universe itself as we know it has time because consciousness needs time to make sense of the universe, to have a concept of on and off, which is the core of logic, which is the core of mathematics.