A cohomological lower bound for the transverse LS category of a foliated manifold

Let $\mathcal{F}$ be a compact Hausdorff foliation on a compact manifold. Let $E_2^{>0,\bullet }=\oplus \{E_2^{p,q}\text{\hspace{0.17em}⁣}:p>0,q\ge 0\}$ be the subalgebra of cohomology classes with positive transverse degree in the $E_2$ term of the spectral sequence of the foliation. We prove that the saturated transverse Lusternik-Schnirelmann category of $\mathcal{F}$ is bounded below by the length of the cup product in $E_2^{>0,\bullet }$. Other cohomological bounds are discussed.