The eventual-to-uniform vanishing conjecture

Let $K$ be a field of characteristic zero, let ${\mathcal A}_K[\xi,z]$ denote the polynomial algebra in the variables $\xi$ and $z$, and let $P(\xi,z)\in {\mathcal A}_K[\xi,z]$. Let $\Lambda$ be the quadratic differential operator occurring in the statement. Assume

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

for all sufficiently large $m$. Eventual-to-uniform vanishing conjecture. Then

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

for every $m\geq 1$. The paper notes that the preceding propositions give a positive answer in a special case; the general assertion is not resolved in the supplied context.