CR analogue of the rigidity theorem for subcritical semilinear equations

Let $M^{2m+1}$ be a closed pseudohermitian manifold with vanishing pseudohermitian torsion $A_{\alpha\beta}=0$ and Webster Ricci tensor satisfying

$$R_{\alpha\overline{\beta}}\geq\frac{m+1}{2}.$$

Suppose a positive function $f$ satisfies

$$-\Delta_b f+\lambda f=f^q,$$

where $\lambda>0$ and $1<q<\left(m+2\right)/m$. If $\lambda(q-1)\leq m/2$, then $f$ is constant unless $q=(m+2)/m$, $\lambda=m^2/4$, and $(M,\theta)$ is isometric to $(\mathbb{S}^{2m+1},\theta_c)$. In the exceptional case,

$$f=c_m\left\lvert\cosh t+(\sinh t)z\cdot\overline{\xi}\right\rvert^{-1/m}$$

for some $t>0$ and $\xi\in\mathbb{S}^{2m+1}$.