Fractional Gaussian noise, the KPSS test, and stationarity

Fractional Gaussian noise (fGn) is characterized by the mean ($\mu$), the standard deviation ($\sigma$), and the Hurst index ($H$). It's my understanding that it is stationary, for the simple reason that the mean, variance, and correlation structure are constant over time.

I also understand that the KPSS test (Kwiatkowski et al., 1992) can be used to check if a time series is stationary. However, the null hypothesis (which, if accepted, corresponds to a decision of 'stationary') is that $\sigma^2 = 0$, versus the alternative that $\sigma^2 > 0$.

In my view, this leads to a contradiction. On one hand, the fGn time series is considered stationary because it has a constant variance, etc. On the other hand, the KPSS test considers a time series to be stationary only if no significant variance is present. Are two different meanings of stationarity involved here? Or have I misunderstood something else?

Looking at the Kwiatkowski paper, they assume the model $$y_t = r_t + \varepsilon_t$$ (forgetting about the trend, i.e. setting $\xi = 0$ in their model). Here $(r_t)$ is a random walk $$r_t = r_{t-1} + u_t$$ with $u_t$ i.i.d. $(0,\sigma_u^2)$ and $(\varepsilon_t)$ is a stationary time series. The assumption that $\sigma_u^2 = 0$ corresponds to $r_t = constant$ in which case $y_t = constant + \varepsilon_t$ is indeed a stationary time series.