Clemens-Griffiths criterion for 3-fold says that if a smooth projective threefold $X/\mathbb C$ is rational, then the intermediate Jacobian $J(X)$ is isomorphic to product of Jacobians $J(C_1)\times \cdots \times J(C_k)$ for some curves $C_i$ as principally polarized abelian varieties.

I'm wondering if the condition is also sufficient (particularly for Fano varieties). In other words, is there a Fano threefold whose intermediate Jacobian is product of Jacobians of curves but it is irrational?

**Edited:** As abx pointed out, Voisin's example and Artin-Mumford's example are not strictly speaking Fano varieties, as they arise from desingularization of nodal quartic double solids, so anti-canonical bundle is not ample on exceptional divisors. It is still interesting to ask if there is such a (smooth) Fano example.