Covering with universally Baire operators

We introduce a covering conjecture and show that it holds below ADR+“Θ is regular”. We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive model containing the reals and ordinals and satisfying ADR+“Θ is regular”. The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of AD++θ0<Θ.