Summary: Your question is tantamount to asking whether mass-energy equivalence has been extended to mass-energy-information (or mass-energy-entropy). As far as I can see, essentially no one in the physics community (outside of a handful of outliers) accepts such an extension. So the answer to your question seems to be, as far it's known, no. I discuss this below, and also explain why entropy and energy are qualitatively different, even though they are both included in free energy expressions.
Anders Sandberg raises the objection that rearranging the letters destroys semantic but not Shannon information. This can be addressed by posing a somewhat different question:
Let's compare two otherwise identical books. Does a book in which the first half are all zeros, and the last half are all ones, have a different mass from an otherwise identical book in which the ones and zeros are randomly arranged? The
The latter would have a higher Shannon entropy, since it requires more information to describe the sequence of numbers in the latter book than the former.
Having Having said this, I believe Anders is correct: Even if the books did have different Shannon entropies, their masses would not differ because of this.
I'll make two related arguments, one based on thermodynamics, and another based on the nature of science generally.
The thermodynamic argument: Mass-energy equivalence applies to, well, energy. Energy is not the same as free energy. Free energies consist of an energy term minus an entropy term (of the form TS). One of the great utilities of a free energy is that it enables us to determine the ability of a system to do work. The more the entropy of the system can increase during a process, the more work (everything else being equal) can be obtained from the system.
So you might ask: Doesn't that mean a lower entropy system has more energy than a higher entropy system? The answer is no. Entropy does not contribute to a system's energy. It is, instead, a measure of the quality of a system's energy—specifically, of how useful (or useless) a system's energy is for doing work.
Sean Carroll has a nice discussion of this on his Preposterous Universe blog:
https://www.preposterousuniverse.com/blog/2010/11/22/using-information-to-extract-energy/
Thus energy and entropy are qualitatively different things. Hence it would require a significant expansion of the concept of mass-energy equivalence to include entropy as a form of energy.
And, as far as I can tell, such an equivalence has not been established. I say this because mass-energy equivalence and entropy-information equivalence are very important in physics. Hence, if mass-entropy (or mass-information) equivalence (which is really what you're asking about) had also been established, this would be a well-known result (because it would connect mass-energy equivalence and entropy-information equivalence!).
Instead, in checking both Google and Google Scholar, I was only able to find a few papers about mass-entropy/mass-information equivalence, including this one:
https://ui.adsabs.harvard.edu/abs/2019AIPA....9i5206V/abstract
Vopson, M.M., 2019. The mass-energy-information equivalence principle. AIP Advances, 9(9), p.095206.
In reading the abstract, and examining the citations, it is clear that mass-energy-information equivalence is, currently, highly speculative.
Hence while is may be possible that the book with lower Shannon entropy (and thus higher free energy) would weigh more, there doesn't appear to be even a small school of physicists that currently accept this.
