Let we have a 3 manifold $M$ with a codimension 1 foliation tangent to $\alpha=0$ for a 1 form $\alpha$. Then $\alpha \wedge d\alpha =0$ then $\exists \beta $ with $d\alpha =\alpha \wedge \beta$. The Godbilon Vey invariant is defined as cohomology class of $\beta \wedge d\beta$.

The conjecture about this invariant is that it is a topological invariant.

What are some updates on this conjecture?