The vanishing conjecture for quadratic differential operators

Let $K$ be a field of characteristic zero, let $P(z)\in {\mathcal A}_K[z]$, and let $\Lambda$ be a quadratic differential operator on ${\mathcal A}_K[z]$. Assume

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

for every integer $m\geq 1$. The vanishing conjecture. Then

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

for all sufficiently large $m$. This conjecture concerns the eventual vanishing of iterated quadratic differential operators and is stated in the paper as equivalent to the Jacobian conjecture. The parser marks the displayed result as resolved, with the cited context attributing the formula to Gurjar and Abhyankar; the exact resolution of the conjectural formulation should be checked.