The Lyapunov genus bound for vector fields with finitely many zeros

Let $X$ be a compact manifold and let $v$ be a smooth vector field that is of gradient type in a neighborhood of its finite zero set $Z(v)$. Let $\mathsf{Lyap}(v)$ be the least cardinality of a finite open cover of $X$ by domains admitting Lyapunov functions for $v$. Lyapunov genus conjecture.

$$\mathsf{Lyap}(v)\leq 2.$$

The bound would extend the equality $\mathsf{Lyap}(v)=1$ for globally gradient-like fields. The source gives no resolution.