Noetherianity conjecture for quadratic-growth algebras with locally nilpotent derivations

Let $k$ be an algebraically closed field of characteristic $0$, and let $A$ be a finitely generated $k$-algebra that is a domain of quadratic growth. Suppose that $A$ does not satisfy a polynomial identity and has a nonzero locally nilpotent derivation. Noetherianity conjecture. Then $A$ is noetherian. The conjecture is motivated by the preceding embedding theorem, which places such algebras inside a noetherian ring of differential operators; the source gives no resolution.