Tom, I finished my homework assignment from you. I read the manuscript carefully and here is my report. First, let me state that I am not an expert in this specific area. If I were asked to review this manuscript for a journal, I would decline on that ground. However, I do know enough to make some qualified judgments.
The authors do a very thorough job of showing data that taken together strongly suggest that they have observed anyon behavior in the devices studied. The strongest data are the discrete jumps whose magnitude and location match what would be expected. Their numerical simulations of the behavior closely match their data.
The manuscript is somewhat long; seven pages, plus two pages containing 69 references (If any of you snickered at the number 69, you may discontinue reading and proceed directly to the Test Board) plus four supplements to amplify their arguments. It is quite common to provide additional details in a supplement (usually published by the journal online) but it is rare to have more than one, let alone four. A paper claiming to have strong evidence of anyons would normally be a candidate for a top tier journal like Science, Nature, or Physical Review Letters, but this is far too long to fit in any of those. I am therefore not sure where they would submit it if they wanted it in a top-tier journal.
A journal editor would send the manuscript to several anonymous reviewers who are experts in the specific subfield. A typical reviewer's response would ask if there is any other explanation besides anyons for the observed behavior. The authors have addressed this possibility to their own satisfaction in their Supplement 4, but one might expect a reviewer to ask why, for erxample, the results from a paper by Joe Shmoe et al. do not provide a non-anyon explanation for the data. More often than not, the anonymous reviewer is in fact Joe Shmoe, and that is why the editor chose him to review it. If the authors provide an adequate response, the paper would be published, perhaps with revisions to address the results of the Schmoe et al. paper. This would also get Joe Shmoe's paper an additional citation.
The above process pretty much summarizes how this manuscript might get published. The bigger question is, so what? Well, anyons have been theoretically predicted, so the unambiguous observation of them would confirm the underlying theory. As for building a quantum computer based on anyons, keep in mind that the measurements were performed at a temperature of 10 milliKelvin, or 0.01 degree Kelvin above absolute zero. Raise the temperature to 0.094 Kelvin (See Figure 4) and the anyon effect almost smears out. Therefore, it would be extremely challenging (but not impossible) to build a usable anyon quantum computer with current technology. Scientific optimism suggests that future advances may make this more plausible.
Anyone who wants to read the manuscript can find it here:
https://arxiv.org/pdf/2006.14115.pdf
The poster who asked above about magnetic monopoles is correct; free magnetic monopoles have never been observed, but they are not forbidden by Maxwell's equations. In fact, adding a term to allow magnetic monopoles would make Maxwell's equations perfectly symmetric, and physicists love symmetry. A fun fact: Paul Dirac showed from relativistic quantum field theory that if a single free magnetic monopole exists somewhere in the universe, then this would force all elementary particles to have quantized electric charge, which is what we observe. This is why physicists keep looking for magnetic monopoles - to explain why all electric charge is quantized - a fundamental unanswered question in itself. This proof requires the existence of a free elementary particle magnetic monopole, not the quasiparticle monopole excitations that the anyons discussed here could provide.
Tom, do I get an A on my homework? Can I take the rest of the summer off? My reply was almost as long as their paper. 8^)