Speck–Strain conjecture on the relativistic thermodynamic coordinate map

Let $n>0$ and $\gamma>0$ denote the particle number density and inverse temperature, respectively, and let $k_B,e,m,c,h$ be the Boltzmann constant, elementary charge, particle mass, speed of light, and Planck constant. Define $S$ and $p$ by

$$S=\mathfrak{H}(n,\gamma)=k_B\ln\left(\frac{4\pi e^4m^3c^2h^{-3}K_2(\gamma)}{n\gamma}e^{\gamma\frac{K_1(\gamma)}{K_2(\gamma)}}\right),$$ $$p=\mathfrak{P}(n,\gamma)=\frac{nmc^2}{\gamma},$$

where $K_1$ and $K_2$ are the modified Bessel functions of the second kind. Speck–Strain's conjecture. The map

$$(n,\gamma)\longmapsto(\mathfrak{H}(n,\gamma),\mathfrak{P}(n,\gamma))$$

is a diffeomorphism of $(0,\infty)\times(0,\infty)$ onto itself. This conjecture concerns the global invertibility of the relativistic thermodynamic variables; the supplied source does not indicate whether it has been resolved.