The vanishing conjecture for Hessian nilpotent polynomials

Let $z=(z_1,z_2,\dots,z_n)$ be free commutative variables, let $D_i=\frac{\partial}{\partial z_i}$, and let

$$\Delta:=\sum_{i=1}^n D_i^2$$

be the Laplace operator on $\mathbb C[z]$. A polynomial $P(z)$ is Hessian nilpotent if its Hessian matrix is nilpotent. Vanishing conjecture. For any Hessian nilpotent polynomial $P(z)$ homogeneous of degree $d=4$, one has

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

when $m\gg0$.

This conjecture is equivalent to the Jacobian conjecture after the homogeneous and symmetric reductions: the relevant symmetric polynomial maps have the form $F(z)=z-\nabla P(z)$, with $P$ homogeneous of degree four and Hessian nilpotent. Its status is not resolved in the supplied source context.