Uniqueness conjecture for nice forms of homogeneous polynomials

Let $p:\mathbb R^n\to\mathbb R$ be a homogeneous polynomial of degree $d$. It is in nice form if

$$p(x)=\sum_i(-1)^{\epsilon_i}x_i^d+q(x),$$

where $q(x)$ has no terms of the form $x_i^d$ or $x_i^{d-1}x_j$, and it is nice if a linear coordinate change brings it to nice form. A polynomial is nonsingular when its associated projective hypersurface is nonsingular.

Uniqueness conjecture. Every nonsingular homogeneous polynomial is nice. For degree greater than $2$, there is precisely one choice of coordinates in which a nice polynomial reaches nice form, up to permutation of coordinates and, if the degree is even, switching signs of coordinates.

This proposes a normal-form and uniqueness statement for nonsingular homogeneous polynomials; the supplied source does not indicate whether it has been resolved.