GARCH_CHECK returns true even when the sum of alpha and beta equals one?
This phenomenon is common with financial time series as the sum of GARCH alpha and beta values approaches one (very close, but not exact). There is no issue with the GARCH_CHECK function implementation as the sum does not equal 1 (slightly smaller). The interpretation of this phenomenon (i.e. sum of alpha and beta approaches 1) is that volatility reverts very slowly to its long-run mean.
This phenomenon had many practitioners and academics advocate an alternative model - IGARCH - to account for the integration in the conditional variance process. The IGARCH brings its own challenges as conditional volatility value does not revert to the long-run mean, thus it can go to infinity.