Conjecture on asymptotic stability of steady shocks for relativistic Burgers flow

Let $M>0$, let $v=v(t,r)\in[-1,1]$ solve the relativistic Burgers equation on the Schwarzschild exterior $r>2M$, and let the initial data be a compactly perturbed steady shock. Asymptotic steady-shock conjecture. The solution converges asymptotically in time to a steady state shock. This proposes late-time stability of steady shocks under compactly supported perturbations; the source presents it as an issue investigated numerically, and no proof or resolution is supplied.