If it is part of the Ilkhanate, I would go as far as to argue that the Copts will actually be favoured. Originally, the Ilkhanate favoured Nestorian Christianity over Islam, and they will likely continue this policy of favouritism for Christians that aren't aligned to a foreign power (i.e. Catholics or Greek Orthodox) over the majority Muslims. This will allow them to establish a 'middle-man' force with the Copts. I doubt that Egypt would remain under the Ilkhanate by the time that the Ilkhans convert to Islam. It is too compact a bloc, and too far from the other Ilkhan centre of power, the Iranian plateau, for the two to remain united long-term. Once the Ilkhans lose Egypt, probably a withdrawal as a result of concurrent rebellions throughout the empire, it is probable that the Copts will seek intervention by the Kingdom of Makuria, which may experience a revival as they occupy the power vacuum. This ends up as an Islam-screw in the short to medium term, as despite the region being thoroughly Muslim in faith, geopolitically the area is dominated by non-Muslims. In the long-term, who knows.