The conjecture on negative transverse curvature for integrable hyperplane distributions

Let $n\geq4$, let $M$ be an $n$-manifold, and let $H$ be an integrable $(n-1)$-plane distribution on $M$. A Riemannian metric is a positive-definite metric $g$ on $M$; the source's function $\chi_{g,\bot H}$ is the scalar quantity associated with $g$ and the orthogonal complement of $H$.

Conjecture on negative transverse curvature. There is a Riemannian metric $g$ on $M$ such that the function $\chi_{g,\bot H}$ is everywhere negative.

The source presents this as an auxiliary conjecture for a possible proof of the esc conjecture. Its status is open in the supplied text; the notation $\chi_{g,\bot H}$ is not defined in the provided context.