Well, the burden of proof doesn't lie with Carroll. Instead, the entire point is that the non-materialist has the burden of evidence:

Given a quantum state of the relevant fields, it accurately predicts how that state will evolve. Skeptics of the claim defended here have the burden of specifying precisely how that equation is to be modified. This would necessarily raise a host of tricky issues, such as conservation of energy and unitary evolution of the wave function.

Otherwise I can rely upon Newton's flaming laser sword; every time you ask about the possibility of non-materialism, I can ask you for the corresponding experiment which opens that possibility. Note that sometimes this is scientifically fruitful, as in the discovery of infrasound leading to many debunkings of hauntings as well as unlocking the secrets of elephant communication. (The more radical position of anti-materialism was conclusively refuted during the colonial era, so we cannot assume that the material world is only hypothetical.)

This is all made stark in Figure 4, p15, which shows that any possible physical force not in the Standard Model would be so weak and subtle as to be undetectable by humans; when a human claims that they are sensitive to such a force, they have incorrectly implicitly assumed that their body is physically capable of interacting with such a force in a perceptible way. The argument goes much like the argument against electrosensitivity: if you really could sense the weak experimental force then you would be constantly sensing the much stronger ambient forces from the outside environment which we can't mute.

A common retort is that quantum states are merely our epistemic knowledge as humans about a fundamentally-unknowable micro-reality below our scale of perception. However, the PBR theorem rules that out by insisting that the quantum wavefunction is ontic. Leifer spent about two years struggling against this result in vain and eventually published Leifer 2014, which both serves as a great overview of the no-go theorems in ontological models and also as an example of how difficult it can be to unlearn previously-accepted beliefs.

By the way, I really hope that you consider synthesizing concepts. As an exercise, Carroll concludes from his premises that:

There is no life after death, as the information in a person’s mind is encoded in the physical configuration of atoms in their body, and there is no physical mechanism for that information to be carried away after death.

But consider the following quote from Strange Loop at the end of Chapter 18, "The Blurry Glow of Human Identity". Remember, Hofstadter is a physicist, arguably as influential as Carroll in quantum theory, and no less of an anti-dualist or materialist. So, as an exercise, synthesize for yourself an understanding of why Hofstadter says:

In the wake of a human being's death, what survives is a set of afterglows, some brighter and some dimmer, in the collective brains of all those who were dearest to them. And when those people in turn pass on, the afterglow becomes extremely faint. And when that outer layer in turn passes into oblivion, then the afterglow is feebler still, and after a while there is nothing left. This slow process of extinction I've just described, though gloomy, is a little less gloomy than the standard view. Because bodily death is so clear, so sharp, and so dramatic, and because we tend to cling to the caged-bird view, death strikes us as instantaneous and absolute, as sharp as a guillotine blade. Our instinct is to believe that the light has all at once gone out altogether. I suggest that this is not the case for human souls, because the essence of a human being — truly unlike the essence of a mosquito or a snake or a bird or a pig — is distributed over many a brain. It takes a couple of generations for a soul to subside, for the flickering to cease, for all the embers to burn out. Although "ashes to ashes, dust to dust" may in the end be true, the transition it describes is not so sharp as we tend to think.