It's just Bell's theorem rebranded, and using a slightly different experiment on entangled particles.
"The vital assumption" of Bell's 1964 argument, in his words, was "that the result $B$ for particle $2$ does not depend on the setting $\vec a$ of the magnet for particle $1$, nor $A$ on $\vec b$." He modeled that by making $A$ a function of $\vec a$ and $λ$, and $B$ a function of $\vec b$ and $λ$, where $λ$ represents everything on which $A$ and $B$ might depend other than $\vec a$ and $\vec b$. He clarifies that $λ$ may contain the entire state of the universe (except $\vec a$ and $\vec b$):
Some might prefer a formulation in which the hidden variables fall into two sets, with $A$ dependent on one and $B$ on the other; this possibility is contained in the above, since $λ$ stands for any number of variables and the dependences thereon of $A$ and $B$ are unrestricted. In a complete physical theory of the type envisaged by Einstein, the hidden variables would have dynamical significance and laws of motion; our $λ$ can then be thought of as initial values of these variables at some suitable instant.
Despite the name, his "hidden" variables are all of the relevant variables, not just unmeasurable non-quantum ones.
Compare that to C & K's MIN assumption:
Assume that the experiments performed by $A$ and $B$ are space-like separated. Then experimenter $B$ can freely choose any one of the 33 particular directions $w$, and $a$’s response is independent of this choice. Similarly and independently, $A$ can freely choose any one of the 40 triples $x, y, z$, and $b$’s response is independent of that choice.
Aside from referring to a different (and rather more complicated) experiment than Bell's, this is the same fundamental assumption, with the same consequences.
C & K also make two named assumptions about quantum mechanics (SPIN and TWIN), while Bell doesn't, but that's only a difference of exposition. Bell could have added certain assumptions about QM and proved a contradiction, but instead, he proved a result from his non-quantum assumption and then observed separately that it's inconsistent with certain assumptions about QM. His assumptions about QM are about as minimal as C & K's; he only needs QM's prediction of the outcome of one simple experiment to be correct. C & K's argument doesn't rule out any model that Bell's argument doesn't, except for models that reproduce QM in C & K's experiment and not in Bell's slightly different one, but those can be falsified by doing Bell's experiment in real life, so they aren't very interesting.
What C & K call "free will" is essentially the negation of what Bell called "superdeterminism", so there's nothing new there either. In fact, in the same 1985 interview where he first used the word "super-deterministic", Bell also called it "the complete absence of free will". It just happened that superdeterminism was the name that stuck. That's probably for the best since, as other people have said, free will as discussed by philosophers is really a moral concept and doesn't have much to do with physics.
The most interesting part of C & K's version of the argument is their experiment, which is similar to the GHZ experiment but uses two three-state particles instead of three two-state particles. It's not new to the paper, which attributes it to Kochen & Specker (1967), Kochen ("1970’s") and Asher Peres (1993).
C & K say
The advantage of the K–S theorem over the Bell theorem is that it leads to an outright contradiction between quantum mechanics and the hidden variable theories for a single spin experiment, whereas the Bell theorem only produces the wrong probabilities for a series of experiments.
I don't buy this argument. First, the quantum prediction is only nonprobabilistic if you can produce entangled pairs with perfect accuracy and align your measurement axes with infinite precision. Accounting for experimental reality makes it probabilistic again. Second, the "outright contradiction" is counterfactual: a local hidden variable theory may match the quantum prediction for the axes you actually choose, but you can prove that there must exist other axes you could have chosen for which it wouldn't have. That's not experimentally testable. All you can do is repeat the experiment with a newly prepared system that's identical as far as you can tell, but may have different values of the hidden variables, so the proof doesn't go through. A local hidden variable theory may pass the single spin experiment or any number of repetitions of it just by getting lucky, so you can only rule it out probabilistically.
For that reason I think C & K's experiment isn't practically or even philosophically better than Bell's, although it's mathematically interesting.
