Mendson–Pereira conjecture for codimension-one foliations on projective space

Let $\mathcal F$ be a codimension one foliation on $\mathbb P^n$, with $n\ge 3$. Let $\Omega^1_{\mathcal F}$ denote its sheaf of foliated $1$-forms. A foliation is rational if it is defined by the fibers of a rational map to a curve. Mendson–Pereira conjecture. If

$$h^0(\mathbb P^n,\Omega^1_{\mathcal F})=0,$$

then $\mathcal F$ is a rational foliation. This conjecture concerns the structure of codimension-one foliations on projective space with no global foliated $1$-forms; its resolution status is not specified in the source.