The finite-dimensionality conjecture for noetherian universal enveloping algebras

Let ${\Bbbk}$ be a field of characteristic $0$, let $L$ be a Lie algebra over ${\Bbbk}$, and let $U(L)$ denote its universal enveloping algebra. Finite-dimensionality conjecture. A Lie algebra $L$ is finite dimensional if and only if the universal enveloping algebra $U(L)$ is noetherian. This conjecture addresses whether the universal enveloping algebra of an infinite-dimensional Lie algebra can be noetherian. The paper proves that the universal enveloping algebra of the Witt algebra, and of its positive part, is not noetherian, but the general equivalence remains open.