The homogeneous vanishing conjecture for Hessian-nilpotent polynomials

Let $P(z)$ be a homogeneous Hessian-nilpotent polynomial of degree $d\geq2$, and define

$$\alpha_{[n,d]}=\frac{1}{d-2}\bigl((d-1)^{n-1}-(d-1)\bigr).$$

Homogeneous vanishing conjecture. Both of the following vanishings should hold:

1. $\Delta^mP^{m+1}=0$ for every $m>\alpha_{[n,d]}$.
2. For every $k\geq1$, $\Delta^mP^{m+k}=0$ for every $m>k\alpha_{[n,d]}$. This is a quantitative reformulation of the homogeneous vanishing conjecture, with an explicit bound derived from inverse-degree estimates; its validity is open in general.