The vanishing conjecture for homogeneous Hessian-nilpotent polynomials

Let $z=(z_1,\ldots,z_n)$ be commuting variables, let

$$\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial z_i^2}$$

be the Laplace operator, and call a formal power series $P(z)$ Hessian nilpotent if its Hessian matrix

$$\left(\frac{\partial^2P}{\partial z_i\partial z_j}\right)_{i,j=1}^n$$

is nilpotent. Vanishing conjecture. For any homogeneous Hessian-nilpotent polynomial $P(z)$ of degree $4$,

$$\Delta^mP^{m+1}=0$$

when $m$ is sufficiently large. This conjecture is equivalent to the Jacobian conjecture after the reductions described in the paper, and is stated as open even though the paper proves it under additional geometric hypotheses.