Thom's conjecture on invariant functions of generic vector fields

Let $M$ be a compact connected manifold, let $X$ be a $C^1$-generic vector field on $M$, and let $f:M\to\mathbb{R}$ be a function that is either $C^1$ or $C^0$. Call $f$ invariant if it is constant along the flow orbits of $X$. Thom's conjecture. Every invariant function $f$ is constant on $M$. This conjecture would yield triviality of the $C^1$-centralizer for generic vector fields with at most finitely many sinks or sources. The source attributes the conjecture to René Thom; its resolution status is not specified here.