Consider a smooth projective threefold $\overline W$, constructed in section 4 of this paper. This threefold is a resolution of singularities of the quotient of a product of a K3 surface and $\mathbb CP^1$ by $(\rho, \psi)$, where $\rho$ is a non-symplectic, non-fixed-point-free involution on the K3 surface and $\psi$ is an involution on $\mathbb CP^1$ (the paper (sec 4) contains more details about the construction).

Then $\overline W$ is simply-connected and it has a smooth K3 surface in its anticanonical system.

Here is my question:

Is $\overline W$ irrational? How can one show it?