Conjectural formulas for the Voronoi degree of generic hypersurfaces

Let $d$ be the degree and let $n$ be the number of variables. For a generic hypersurface in $\mathbb{R}^n$, its Voronoi degree is the degree of the corresponding Voronoi ideal. For a generic homogeneous polynomial, its zero set is a cone.

Voronoi degree conjecture. The Voronoi degree of a generic hypersurface of degree $d$ in $\mathbb{R}^n$ equals

$$(d-1)^n + 3(d-1)^{n-1} + \tfrac{4}{d-2}\bigl((d-1)^{n-1}-1\bigr) - 3n.$$

The Voronoi degree of the cone of a generic homogeneous polynomial of degree $d$ in $\mathbb{R}^n$ is

$$2(d-1)^{n-1} + \tfrac{4}{d-2}\bigl((d-1)^{n-1}-1\bigr) - 3n + 2.$$

These formulas summarize computational experiments for generic inhomogeneous and homogeneous hypersurfaces. The source reports that the relevant Voronoi ideals were zero-dimensional, and in fact maximal ideals over $\mathbb{Q}$ in the sampled cases; the formulas remain conjectural.