Boundedness of elliptic Calabi-Yau varieties with a rational section

We show that for each fixed dimension $d\geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical class is bounded and the elliptic fibration admits a rational section. This case builds on an analogous boundedness result for the set of rationally connected log Calabi-Yau pairs with bounded torsion index. In dimension $3$, we prove the more general statement that the set of $\epsilon$-lc pairs $(X,B)$ with $-(K_X +B)$ nef and rationally connected $X$ is bounded up to isomorphism in codimension one.