Birational boundedness of low dimensional elliptic Calabi-Yau varieties with a section

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds $Y\to X$ with a rational section, provided that $\dim (Y)\le 5$ and $Y$ is not of product-type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of klt pairs $(X,\mathrm{\Delta })$ with $K_X+\mathrm{\Delta }$ numerically trivial and not of product-type, in dimension at most $4$.