The Dicksonianity conjecture for polynomial-growth graded Lie algebras

Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let $\mathfrak g$ be a graded Lie $\Bbbk$-algebra of polynomial growth, also called finite growth. Suppose that $\mathfrak g$ has the ascending chain condition on Lie ideals.

Dicksonianity conjecture. Then $\mathfrak g$ is Dicksonian.

If true, this would imply, by the paper's main theorem, that $S(\mathfrak g)$ has the ascending chain condition on radical Poisson ideals. The source presents this as a short-term goal and gives no resolution, so it remains open.