A unique property of a dynamically generated quantum spin Hall state are Goldstone modes that correspond to the long-wavelength fluctuations of the spin-orbit coupling order parameter whose topological Skyrmion excitations carry charge 2$e$. Within the model considered here, upon varying the chemical potential, we observe two transitions: An s-wave superconducting order parameter develops at a critical chemical potential $mu_{c1}$, corresponding to the excitation gap of pairs of fermions, and at $mu_{c2}$ the SO(3) order parameter of the quantum spin Hall state vanishes. Using negative-sign-free, large-scale quantum Monte Carlo simulations, we show that $mu_{c1}=mu_{c2}$ within our accuracy -- we can resolve dopings away from half filling down to $delta = 0.0017$. The length scale associated with the fluctuations of the quantum spin Hall order parameter grows down to our lowest doping, suggesting either a continuous or a weakly first-order transition. Contrary to mean-field expectations, the doping versus chemical potential curve is not linear, indicating a dynamical critical exponent $z > 2$ if the transition is continuous.
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