How accurately does Hawking emission survive modified short-distance dispersion?

Take a $1+1\mathrm{D}$ field with dispersion $(\omega - v(x) k)^2 = c^2 k^2 (1 + s k^2/k_d^2)$, $s = +1$ (superluminal) or $-1$ (subluminal), on a stationary flow $v(x)$ with a sonic horizon of surface gravity $\kappa = \left|d(v - c)/dx\right|$ at the horizon. Compute the emitted spectrum $n(\omega)$ and its deviation from the Planck law at $T_H = \hbar \kappa/(2 \pi)$, including the cutoff frequency $\omega_{\max}$ and the greybody factor. An answer is a proven bound on $\left|n(\omega) - n_{\mathrm{Planck}}(\omega)\right|$ as a function of $\kappa/(c k_d)$ and the width $D$ of the near-horizon region, valid also for steep profiles with $\kappa \sim c k_d$.

An analytic $S$-matrix error bound, checked against exact numerical mode solutions over the full range of $\kappa/(c k_d)$ and $D k_d$.