The vanishing conjecture for homogeneous Hessian-nilpotent quartics

Let $z=(z_1,z_2,\dots,z_n)$, let $D_i=\frac{\partial}{\partial z_i}$, and let $\Delta_n=\sum_{i=1}^n D_i^2$ be the Laplace operator on the polynomial algebra ${\mathcal A}_n=\mathbb C[z_1,\dots,z_n]$. A polynomial $P(z)\in{\mathcal A}_n$ is Hessian nilpotent when its Hessian matrix $(\frac{\partial^2P}{\partial z_i\partial z_j})_{n\times n}$ is nilpotent. The vanishing conjecture. For any homogeneous Hessian-nilpotent polynomial $P(z)\in\mathcal A_n$ of degree $4$, one has

$$\Delta_n^m P^{m+1}(z)=0$$

when $m\gg0$. This vanishing conjecture is a further reduction of the Jacobian conjecture; its status is not specified as resolved in the source.