Strong Wilhelm's conjecture for positively curved Riemannian foliations

Let $M$ be a compact manifold with positive curvature, and let $\cal F$ be a Riemannian foliation on $M$. A horizontal vector is a vector tangent to the horizontal distribution of the foliation, and $A^*_X$ denotes the corresponding adjoint O'Neill tensor. Strong Wilhelm's conjecture. There is a horizontal vector $X$ such that $A^*_X$ is injective. The conjecture is motivated by results suggesting that positively curved foliations carry fat objects; the supplied text gives no resolution.