Large-parameter scaling of bounded amoeba-complement volumes

Let $P(\boldsymbol{z})=k-p(\boldsymbol{z})$ be a Laurent polynomial in $\boldsymbol{z}\in\mathbb{C}^n$, and let $\mathcal{A}(P)$ be its amoeba. Write $V_\textup{bdd}$ for the volume of a bounded complementary region of $\mathcal{A}(P)$. Bounded-volume scaling conjecture. In the large-$k$ limit,

$$V_\textup{bdd}\sim\log^n k.$$

The two-dimensional analysis suggests logarithmic growth of hole areas, and the source proposes the analogous scaling in every dimension $n$; it does not establish the general claim.