A hypergeometric proof that ${\sf Iso}$ is bijective

We provide a short and elementary proof of the main technical result of the recent article "On the uniqueness of Clifford torus with prescribed isoperimetric ratio" by Thomas Yu and Jingmin Chen. The key of the new proof is an explicit expression of the central function (${\sf Iso}$, to be proved bijective) as a quotient of Gaussian hypergeometric functions.