Two recent studies (Denkov et al., Nature 2015, 528, 392 and Guttman et al. Proc. Natl. Acad. Sci. U.S.A.2016, 113, 493) demonstrated that micrometer-sized n-alkane drops, dispersed in aqueous surfactant solutions, can break their spherical symmetry upon cooling and self-shape into a variety of regular shapes, such as fluid polyhedra, platelet-shaped hexagons, triangles, rhomboids, toroids, and submicrometer-diameter fibers. In the first study, the observed phenomenon was explained by a mechanism involving the formation of interfacial multilayer of self-assembled alkane molecules in the so-called rotator phases, templated by the frozen surfactant adsorption layer. Such phases are known to form in alkane droplets under similar conditions and are sufficiently strong to deform the droplets against the capillary pressure of a finite interfacial tension of several mN/m. The authors of the second study proposed a different explanation, namely, that the oil-water interfacial tension becomes ultralow upon cooling, which allows for surface extension and drop deformation at negligible energy penalty. To reveal which of these mechanisms is operative, we measure in the current study the temperature dependence of the interfacial tensions of several systems undergoing such drop-shape transitions. Our results unambiguously show that drop self-shaping is not related to ultralow oil-water interfacial tension, as proposed by Guttmann et al. These results support the mechanism proposed by Denkov et al., which implies that the large bending moment, required to deform an oil-water interface with an interfacial tension of 5 to 10 mN/m, is generated by an interfacial multilayer of self-assembled alkane molecules.