Ascending-chain conjecture for two-sided ideals of the positive Witt enveloping algebra

Let $W_+$ be the positive Witt algebra over a field of characteristic zero, and let ${\rm U}(W_+)$ be its universal enveloping algebra. The two-sided ideals of ${\rm U}(W_+)$ satisfy the ascending chain condition: every strictly ascending chain of two-sided ideals is finite.

The conjecture would give a strong finiteness property for the two-sided ideal lattice of ${\rm U}(W_+)$, despite the algebra being neither left nor right noetherian. It is proved in the paper for certain classes of ideals, but the assertion for all two-sided ideals remains open.