Non-vanishing perturbation conjecture for non-closed 1-forms

Let $M$ be a closed manifold with zero Euler characteristic, and let $\alpha$ be a non-closed differential $1$-form on $M$. Non-vanishing perturbation conjecture. There exists a smooth function $f:Meto {\bf R}$ such that the $1$-form

$$df+\alpha$$

has no zeroes on $M$. This is stated as a more general conjecture from which the cylinder conjecture follows; the source gives no resolution.