Mixed-volume asymptotic conjecture for amoeba contours

Let $V_d=\{f_{1,d}=\cdots=f_{r,d}=0\}\subset(\mathbb C^\ast)^n$ be the complete intersection under consideration, and let $\Delta_{i,d}$ be the Newton polytope of $f_{i,d}$. Set $k=n-r$, and let $\Theta_{1,d},\ldots,\Theta_{k,d}$ be the Newton polytopes associated with the logarithmic Plücker coordinates. Write $MV$ for the corresponding mixed volume. Mixed-volume asymptotic conjecture. The contour degree satisfies

$$\mathbb R\deg(\mathcal C\mathscr A_{V_d})=O\left(MV(\Delta_{1,d},\dots,\Delta_{r,d},\Theta_{1,d},\dots,\Theta_{k,d})\right).$$

This refines the degree-growth prediction by expressing the bound in toric terms. Bernstein-type reasoning and the support containment of logarithmic derivatives provide evidence, but the supplied text does not state a resolution; the conjecture remains open.