We extend recent work on a leakage-protected, adiabatic entangling gate for exchange-only spin qubits [Doherty and Wardrop, PRL 111, 050503 (2013)] by adapting to a setting where single spins are not assumed to be polarized on preparation. Previous gate constructions do not function correctly when gauge spins are uninitialized, because the entangling gate has different, non-trivial action in different gauge subspaces. Our construction inherits many of the desirable features of the previous work while addressing the gauge-dependent behavior. Using numerical simulation, we show that the resulting gate implements the same logical operation in both gauge subspaces to first order in perturbation theory, and second-order terms introduce an error that decreases quadratically in the duration of the gate. We add $1/f$ charge noise to voltages modulating exchange in this model, which introduces errors that increase with gate time, to show that there is an optimal gate duration for a given set of device parameters.
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