The kinetic equation of state auto-diffeomorphism conjecture

Let $n$ be the proper number density, $\theta$ the temperature, $\beta=m_0c^2/(k_B\theta)$ the dimensionless inverse-temperature parameter, $p$ the pressure, $\rho$ the proper energy density, and $\eta$ the entropy per particle. Let $K_1$ and $K_2$ be the modified Bessel functions appearing in the kinetic equation of state

$$p=k_Bn\theta=m_0c^2\frac{n}{\beta},$$ $$\rho=m_0c^2n\frac{K_1(\beta)}{K_2(\beta)}+3p,$$ $$n=4\pi e^4m_0^3c^3h^{-3}\exp\left(\frac{-\eta}{k_B}\right)\frac{K_2(\beta)}{\beta}\exp\left(\beta\frac{K_1(\beta)}{K_2(\beta)}\right).$$

Kinetic equation of state auto-diffeomorphism conjecture. The map $(n,\beta)\leftrightarrow(\eta,\rho)$ given implicitly by these equations is an auto-diffeomorphism of $]0,\infty[\times]0,\infty[$.

This is one of the global properties needed to formulate the relativistic Euler system in thermodynamic variables. It was established in certain temperature regimes in the cited prior work, but the statement was conjectured to hold for all positive $n$ and $\beta$.