Ideals in the enveloping algebra of the positive Witt algebra

Let $W_+$ be the {\em positive Witt algebra}, which has a $\CC$-basis $\{e_n: n \in \ZZ_{\geq 1}\}$, with Lie bracket $[ e_i, e_j] = (j-i) e_{i+j}$.
We study the two-sided ideal structure of the universal enveloping algebra $\U(W_+)$ of $W_+$.
We show that if $I$ is a (two-sided) ideal of $\U(W_+)$ generated by quadratic expressions in the $e_i$, then $\U(W_+)/I$ has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition.
We conjecture that analogous facts hold for arbitrary ideals of $\U(W_+)$, and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra ${\rm S}(W_+)$.