The A-cone conjecture for Morse-Novikov numbers of links

Let $L$ be a link in $S^3$. For a natural number $k$, define

$$\Gamma_k=\{\xi\in H^1(E(L),{\mathbb{Q}})\mid \xi\text{ is regular and }{\mathcal M}{\mathcal N}(E(L),\xi)=k\}.$$

Here a subset $A$ of a finite-dimensional ${\mathbb{Q}}$-vector space $H$ is an A-cone if, for some finite family of hyperplanes $S_k$ in $H$, the set $A\setminus\bigcup_k S_k$ is a finite union of basic convex cones, where a basic convex cone is a finite intersection of open half-spaces.

A-cone conjecture. For every natural number $k$, the set $\Gamma_k$ is an A-cone.

The conjecture predicts that the Morse-Novikov number is organized by finitely many convex conical regions, up to finitely many hyperplanes, as the cohomology class varies. It is motivated by the corresponding established A-cone result for the Novikov-homology number, but the source gives no resolution of this stronger assertion.