The Newton non-degenerate homaloidal hypersurface conjecture

Let $f$ be a Newton non-degenerate homogeneous polynomial of degree $d\geq 3$ defining a homaloidal hypersurface. Let $D_0,\dots,D_n$ be the polytopes associated with $f$ in Proposition 4.9: if $E_0^d,\dots,E_n^d$ are the facets of $C_d$ and

$$D_i=\operatorname{Conv}\bigl((\mathbf P\ast\mathbf\Delta^\circ)\setminus E_i^d\bigr)\subset C_d,$$

then these are the polytopes whose mixed volume gives the polar degree. Homaloidal hypersurface conjecture. There is a renumbering of $D_0,\dots,D_n$ such that, for every $0\leq i\leq n$, the quotient

$$D_i/(D_0+\dots+D_{i-1})$$

is a segment of lattice length $1$. This gives a proposed combinatorial characterization of Newton non-degenerate homaloidal hypersurfaces; the authors verify it for the examples cited from Huh and report no counterexamples, but the general assertion remains open.