The vanishing conjecture for differential operators with constant coefficients

Let $z=(z_1,, z_n)$ be $n$ commuting variables, let $\mathbb C[z]$ be the polynomial algebra, and write $\partial=(\partial_1,\dots,\partial_n)$ with $\partial_i=\partial/\partial z_i$. Let $P(z)\in\mathbb C[z]$ and let $\Lambda=\Lambda(\partial)$ for some $\Lambda(\xi)\in\mathbb C[\xi]$. Vanishing conjecture. If

$$\Lambda^m(P^m)=0$$

for every $m\geq 1$, then

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

for all sufficiently large $m$. This conjecture concerns the eventual vanishing of iterated constant-coefficient differential operators and is related to the Jacobian conjecture. The paper proves four cases, but the general statement remains unresolved in the supplied text.