Causal self-consistency of the Blandford--McKee self-similar solution

The Blandford--McKee (BM) solution describes the ultra-relativistic self-similar flow behind a strong spherical blast wave propagating into an external density profile $\propto r^{-k}$, where $k<4$ and $r$ is the distance from the center, but it applies only to the hot shell adjacent to the shock, s