What is the difference between "dhann" (conjecture) and "shakk" (doubt)? And what is the ruling on "dhann" (conjecture) regarding Allah, Islam, and the religion?

The terminological difference between al-ẓann (conjecture/preponderant belief) and al-shakk (doubt) is that al-ẓann refers to the probable possibility, while al-shakk refers to the equality of two possibilities without preferring one over the other, as stated by Imam al-Haramayn. In language, there is sometimes no difference between them, and al-ẓann may even come to mean certainty (yaqīn). What requires taṣdīq (assent/affirmation) for the soundness of faith is not benefited by al-shakk or al-ẓann; rather, it demands firm conviction. As for matters of ijtihād (independent reasoning), preponderant ẓann ( al-ẓann al-ghālib) suffices.