Lo! As for those who sunder their religion and become schismatics, no concern at all hast thou with them. Their case will go to Allah, Who then will tell them what they used to do.
Times are computed by method; your mosque's timetable governs the adhan. Every text carries its source and grade. Nothing here is a fatwa, a prediction, or a substitute for a doctor: your own scholar decides what is contested, and the tradition itself says the practice is alongside treatment, never instead of it.