However, 'deal with the challenges' means 'be comfortable with the demands'.
'deal with' in this context roughly means 'work with'. In other words, can the Islamic rulers be comfortable with adapting their practices to a situation where women want more equality? 'Deal' here does not have its usual English meanng of 'barter' - rather, it means 'I do not like it, but I can come to terms with it'. The British ex-Prime Minister Mrs. Thatcher used it this way when she said of Mikhael Gorbachev "This is a man we can deal with."
'Challenges' here does not refer to the actions of the women's movement. It means that the Islamic rulers must alter their behaviour to accommodate the demands of the women's movement - this is a 'challenge' to them. Can they respond?