The non-noetherianity conjecture for infinite-dimensional subalgebras of the Witt algebra

Let $W_{\geq -1}$ be the Lie algebra of polynomial vector fields on the affine line with degrees at least $-1$, and let $U(-)$ denote the universal enveloping algebra. Non-noetherianity conjecture. If $\mathfrak{g}$ is an infinite-dimensional Lie subalgebra of $W_{\geq -1}$, then $U(\mathfrak{g})$ is not noetherian. The conjecture is proved for graded subalgebras and for subalgebras of finite codimension, but remains open for arbitrary infinite-dimensional subalgebras of infinite codimension.