Liquid and gas contributions determined by amoeba geometry

Let $P(z,w)=k-p(z,w)$ be a Newton polynomial with fixed $p(z,w)$. Let $m_l(P)$ and $m_g(P)$ denote the liquid and gas contributions to the Mahler measure, and let $\mathcal{A}(P)$ be the amoeba, with area $A(\mathcal{A})$ and total hole area $A_h$. Phase-contribution conjecture. The liquid contribution $m_l(P)$ is determined only by $A(\mathcal{A})$, while the gas contribution $m_g(P)$ is determined only by $A_h$. This proposed decomposition aims to relate the two phase contributions along the Mahler flow to the corresponding amoeba geometry; the source does not prove the asserted dependence.