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The hypothesis presented in this manuscript is intuitively attractive, but poorly supported. At the very least, the manuscript would benefit from a more extensive review of the relevant literature on this subject, including Haack, S.C. (1986). A Thermal Model of the Sailback Pelycosaur. Paleobiology. 12(4):450-458. The work should be acknowledged, and (where necessary) refuted. Also J.B. Bailey's work in the 1990s on sail backs deserves to be mentioned. One may disagree with Bailey's highly unorthodox (and controversial) hypothesis that Spinosaurus (and other tall-spined dinosaurs) had bison-like humps. I certainly disagree. But the study is worth reading because it highlights the morphological distinctions in the shape of the neural spines between different groups. There is more to a 'sail back' than simply the size of the sail.
Within dinosaurs, tall neural spines are not unique to Spinosaurus. They are also found in a number of dinosaur taxa, including the ornithopod Ouranosaurus - which lived around the same time and place as Spinosaurus. Also, elongation of spines to various degrees is also found in Acrocanthosaurus, Concavenator, Deinocheirus, Altispinax, Amargasaurus, Morelladon, etc. Outside the Dinosauria, there are sail-backed archosauriforms such as Ctenosauriscus, Arizonasaurus, Hypselorhachis. Lotosaurus, and Xilousuchus. Some of these show extreme development of the sail. There is no evidence that all of these 'sail-backed' (or at least tall-spined) animals were amphibious or semi-aquatic.
Thus, it is difficult to support the hypothesis that evolving a sail-backed morphology is associated with the lifestyle proposed here ("thermoregulatory function, warming the animals, otherwise submerged in the water"). If this interpretation holds for Spinosaurus and sphenacodontids, why not other sail-backed vertebrates? The author's defense is "so the comparison is restricted here to these two clades only" on account of this feature (sail back) being most extremely developed in the two groups, and because they apparently "share a piscivorous and possibly semiaquatic mode of life". But certain other taxa (non-spinosaurids and non-sphenacodontids) have quite prominent sails - like Edaphosaurus (mentioned in passing in the manuscript), as well as some of the taxa I mentioned above. There is no justification for a comparison of spinosaurids and sphenacodontids that excludes other tall-spined taxa.
Regarding "The tall sails of spinosaurids and sphenacodontids would also make quite a big shade on the water surface. It is hypothesized here that this shade might attract fish and enable the animals in question to prey in a way similar to the Recent black heron"... Was the neck of Spinosaurus long enough or flexible enough to reach around, for the jaws to grab any fish that were 'shaded' by the sail? I doubt that it was.
Finally, it's a shame that David Peters uses his 'review' to spruik his own pseudoscientific website. As a rule, Peters' 'work' is best ignored.