Erosion rates for a relativistic electron beam propagating through a low‐density plasma channel in the ion‐focused regime are computed via axisymmetric particle simulation. Steady‐state inductive (Ohmic) erosion rates are found that are in quantitative agreement with new theoretical results, which correctly account for the relativistic dynamics of electrons at the beam head. In cases where the finite emittance of the beam is expected to be the dominant loss mechanism, the beam does not develop steady‐state erosion, as suggested by previous analyses. Asymptotically, such a beam is characterized by a low‐emittance population at the beam head, a long rise length, and an erosion rate that tends toward zero.