The homological independence conjecture for toric cycles

Let $V=P^{-1}(0)\cap(\mathbb{C}^{\times})^n$ be an algebraic hypersurface in the complex torus $(\mathbb{C}^{\times})^n$, and let $\mathscr{A}_V$ be its amoeba under the logarithmic map

$$\mathop{\mathrm{Log}}\nolimits(z)=(\log|z_1|,\ldots,\log|z_n|).$$

For each connected component $E_\nu$ of $\mathbb{R}^n\setminus\mathscr{A}_V$ corresponding to an integer point $\nu$ of the Newton polytope $\Delta_P$, choose $x\in E_\nu$ and define the toric cycle

$$\Gamma_\nu=\mathop{\mathrm{Log}}\nolimits^{-1}(x)\subset(\mathbb{C}^{\times})^n\setminus V.$$

Homological independence conjecture. The toric cycles $\Gamma_\nu$ constitute a homologically independent family in the homology group $H_n((\mathbb{C}^{\times})^n\setminus V)$.

This conjecture concerns the homology of complements of algebraic hypersurfaces and is motivated by the role of toric cycles in amoeba theory and multidimensional residue theory. The source notes that independence was previously proved for cycles associated with vertices of the Newton polytope, while the asserted independence for all toric cycles remains unresolved here.