The product-cover conjecture for foliations with numerically trivial tangent bundle

Let $\mathsf{X}$ be a compact Kähler manifold, and let $\mathcal{F}$ be a regular foliation of $\mathsf{X}$ with $c_1(\mathcal{T}_{\mathcal{F}})=0$.

Product-cover conjecture. There exist possibly noncompact Kähler manifolds $\mathsf{Y}$ and $\mathsf{Z}$, with $\omega_\mathsf{Y}$ trivial, and a covering map

$$f:\mathsf{Y}\times\mathsf{Z}\to\mathsf{X}$$

such that $f^{-1}\mathcal{F}$ is induced by the projection to $\mathsf{Z}$.

This formulation combines the conjectured tangent-bundle splitting with Beauville's conjecture on universal-cover products. It would describe the foliation after a possibly infinite cover, but remains open.