Is the fast of the villagers valid if the muezzins delay the Fajr Adhan by six minutes, causing them to break their fast after the beginning of the fasting time?

Muslims must adhere to the correct timings. The determinant for a fasting person to stop eating is the true dawn (al-fajr al-sadiq). If the people of an area ate after the adhan, believing that the dawn had not yet arrived, the majority of scholars oblige them to make up those days if it becomes certain that dawn had indeed broken. However, if there is doubt, making up the days is not required. Shaykh al-Islam Ibn Taymiyyah holds the view that making up the days is not necessary in this case. The more cautious approach is to make them up if the breaking of dawn becomes certain. A slight difference in timing does not necessitate certainty that one consumed something that breaks the fast during the time of dawn, especially with the disagreement regarding the calendar's alignment with the true dawn. Therefore, the fast is mostly valid, and it does not appear that making up the days is necessary unless it is certain that the eating occurred after dawn had clearly broken. They must be instructed to adjust the adhan time if it is confirmed that they are calling the adhan after the correct time.