Birational boundedness of rationally connected Calabi–Yau 3-folds

We prove that rationally connected Calabi–Yau 3-folds with Kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ϵ-CY type form a birationally bounded family for ϵ>0. Moreover, we show that the set of ϵ-lc log Calabi–Yau pairs (X,B) with coefficients of B bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi–Yau 3-folds with mld bounded away from 1 are bounded modulo flops.