Does the prayer of a congregant become invalid if their intention in reciting Al-Fatiha when the Imam starts it is to read it, not to alert the Imam or correct his mistake?

The issue is a matter of dispute among scholars. The Hanafis are of the view that the intention of prompting (fatḥ) is obligatory, not the intention of recitation (qirā’ah), because, according to them, recitation by the follower is prohibited. The Shafi'is, however, mandated the intention of recitation, and prayers are invalid without it according to them. As for the Hanbali jurists, they permitted prompting the Imam without addressing what the one who prompts intends, and their unqualified allowance implies the validity of the prayer for whoever intended any of that.