The Newton-polytope determination conjecture for points on hypersurfaces

Let $C_f$ be the hypersurface defined by a polynomial $f$, let $H_f$ be its Newton polytope, let $G(H)$ be the greatest common divisor of all coordinates of all elements of $H$, and let $cmathcal E(C_f)$ and $cmathcal E(H)$ be the finite sets of integer multiples introduced in the paper. Newton-polytope determination conjecture. For almost all $f$ with $H_f=H$, where all coefficients vary and the polynomials are ordered by coefficient size, we have

$$G(C_f)=G(H)\quad\text{and}\quad cmathcal E(C_f)=cmathcal E(H).$$

Equivalently, $D(C_f)^{\mathrm{inf}}$ is typically entirely determined by $H_f$. The claim formalizes the idea that the degrees occurring infinitely often should usually depend only on the Newton polytope, although the supplied text does not state its resolution.