  We study spectral gaps of the one-dimensional totally asymmetric simple
exclusion process (TASEP) with open boundaries in the maximal current phase.
Earlier results for the model with periodic boundaries suggest that the gaps
contributing to the universal KPZ regime may be understood as points on an
infinite genus Riemann surface built from a parametric representation of the
cumulant generating function of the current. We perform explicit analytic
continuations from the known large deviations of the current for open TASEP,
and confirm the results for the gaps by an exact Bethe ansatz calculation, with
additional checks using high precision extrapolation numerics.
