The conjecture on differentiability of the Ising magnetization away from criticality

Let $G$ be an infinite Cayley graph. The magnetization $m^+(\beta,h)$ is the plus-state magnetization, with $\beta,h\in[0,\infty)$ and $\beta_c$ the critical inverse temperature. Differentiability conjecture. The function $m^+(\beta,h)$ is infinitely differentiable as a function of $(\beta,h)$ on

$$[0,\infty)^2\setminus\{(\beta_c,0)$$

. The conjecture concerns smoothness away from the critical point and is most interesting when $h=0$ and $\beta>\beta_c$. The source does not provide a resolution, so its status remains open.