Suppose we have a compact three-manifold $M$, a codimension-one foliation $F$ of $M$, and a one-form on $\alpha$, chosen so that $F$ is tangent to $\alpha = 0$. We deduce that $\alpha \wedge d\alpha = 0$. Also, there is some one-form $\beta$ so that $d\alpha = \alpha \wedge \beta$.

The Godbillon-Vey invariant of $F$ is defined as the cohomology class of $\beta \wedge d\beta$. This class is independent of the choices made (at least up to sign).

Then we have the following:

Conjecture: The Godbillon-Vey invariant is a topological invariant.

That is, two topologically equivalent foliations have the same Godbillon-Vey class.

What are some updates on this conjecture?