Are $E>B$ regions essential for particle injection in relativistic magnetic reconnection?

During relativistic magnetic reconnection, X-points ($E>B$ regions at vanishing guide field) have been proposed to play an essential role in injecting particles into the nonthermal population. Using fully kinetic particle-in-cell simulations at a range of magnetizations $\sigma \equiv B_0^2\mathrm{/}(4\pi n_0{m}_e{c}^2)=12$–$800$, we study in detail how the particle energy gains up to the injection threshold ${\gamma }_{\mathrm{i}\mathrm{n}\mathrm{j}}=\sigma \mathrm{/}4$ are partitioned into work done inside $E>B$ regions (unmagnetized), within one gyration after exit (re-magnetizing), and at all other times (magnetized). The magnetized channel supplies the largest part of the energy gain for $\sim 95\mathrm{\%}$ of injected tracers at $\sigma =12$ and $\sim 65\mathrm{\%}$ at $\sigma =800$. Most of the tracers in this magnetized majority never enter an $E>B$ region before injection. We also compare these injection shares with the earlier studies that proposed an essential role for $E>B$ regions. We conclude that $E>B$ regions are neither necessary for injection nor the dominant direct contributors to pre-injection work by injection share in two-dimensional, nonradiative pair-plasma reconnection at vanishing guide field.