Geometric representability of 1-cycles on rationally connected threefolds

We prove that for any rationally connected threefold $X$, there exists a smooth projective surface $S$ and a family of $1$-cycles on $X$ parameterized by $S$, inducing an Abel-Jacobi isomorphism $\mathrm{A}\mathrm{l}\mathrm{b}(S)\cong J^3(X)$. This statement was previously known for some classes of smooth Fano threefolds.