The bordered-manifold Hopf brushing conjecture

Let $W$ be a compact bordered manifold, first in the $C^{\infty}$ and then in the $C^0$ category. Let $c7(W)$ denote its Euler characteristic, and let $M_i$ be the components of its boundary.

Bordered-manifold Hopf brushing conjecture. Let $W$ be a compact bordered manifold (first $C^{\infty}$ and then $C^0$). Assume that $c7(W)=0$, and when $W$ is odd-dimensional that each components $M_i$ of $\partial W$ has $c7(M_i)=0$. Then $W$ admits a brushing.

The boundary condition is motivated by the obstruction that a flow preserves the boundary and by Poincare duality. The source asks whether these conditions suffice, without providing a resolution.