We present a series of unprecedently long 2D3V PIC simulations of unmagnetized relativistic $e^{-}{e}^{+}$-pair shocks performed in a front-comoving frame. By implementing a moving-wall boundary condition in the downstream together with continuous injection at the upstream boundary, we maintain a fixed simulation domain size, opening the way to perform substantially longer simulations. Our longest runs extend beyond $100000{\omega }_p^{-1}$, exceeding the duration of the previously published simulations by a factor of several. Across a diverse set of simulations — varying upstream/downstream lengths, transverse sizes, and particle-per-cell counts — we find strong evidence that the shock, modelled for a long-time in the front-comoving frame, approaches an asymptotic, time-independent state. In the downstream region, the steady state depends only on the upstream temperature at the injection boundary and remains the same if we change the downstream and upstream lengths. The upstream precursor evolves slower and retains a dependence on the simulation’s upstream length, that may be of minor observational consequence, since radiation from astrophysical shocks predominantly originates from the downstream region. Our simulations also indicate that Fermi-type acceleration is limited in energy and a true power-law tail never forms. Another important finding is that the downstream magnetic field has a soliton-like structure, where individual magnetic domains evolve independently, each comprising a compact, highly magnetized core embedded within an extended, weakly magnetized region. The magnetic-field distribution around the centers of these spots has approximately Lorentzian profile.