What is the legal ruling on a man and his daughter drinking after the Fajr call to prayer, based on a shaykh's statement that the call to prayer occurs 13 to 23 minutes before the true Fajr?

There is no blame on someone who eats or drinks, believing that the time for True Dawn (Fajr As-Sadiq) has not yet begun, or who is in doubt about its arrival. This is because the original state is the continuation of night, and Allah has commanded eating and drinking until "the white thread of dawn becomes distinct to you from the black thread [of night]."