Foliated Fujita conjectures for adjoint line bundles

Let $\mathcal{F}$ be a regular foliation of rank $p$ on a smooth projective variety $X$, and let $L$ be an ample line bundle on $X$. Foliated Fujita conjectures. The line bundle

$$K_{\mathcal{F}}+(p+1)L$$

should be globally generated, and the line bundle

$$K_{\mathcal{F}}+(p+2)L$$

should be very ample. These are foliated analogues of Fujita's freeness and very ampleness conjectures for adjoint line bundles; their status is not resolved in the supplied text.