Before revision:  If the protein explores the native-like states for a time much longer than the time to overcome the free energy barrier,  the resulting FPT distribution may be approximately single-exponential, as is expected for two-state folders, but the mean FPT (MFPT) to reach the native state will be determined by the time to find the native state among the native-like ones.
After revision:  If the protein explores the native-like states for a time much longer than the time to overcome the free energy barrier, the resulting FPT distribution to reach the native state remains close to exponential but the mean FPT (MFPT) to reach the native state will be determined by the time to find the native state among the native-like ones.