The Weyl-algebra factorization conjecture for the negative Witt algebra

Let $\W{-1}$ denote the negative Witt algebra, let $\Ua(\W{-1})$ be its enveloping algebra, and let $A_k$ denote the relevant $k$-th Weyl-algebra target. For each $n$, write $\Psi_n=\Psi^{\W{-1}}_n$. Weyl-algebra factorization conjecture. For any ring homomorphism $\varphi:\Ua(\W{-1})\to A_k$, there exists $\ell<k$ and $\mathbf{n}=(n_1,\dots,n_\ell)\in\mathbb{N}^\ell$ such that $\varphi$ factors through $\Psi_{\mathbf{n}}=(\Psi_{n_1}\otimes\dots\otimes\Psi_{n_\ell})\circ\Delta^\ell$; that is, there exists a homomorphism $\overline{\varphi}$ making the corresponding factorization diagram commute. The analogous assertion holds for $W$, with $A_k$, $A_\ell$, and $\Psi^{\W{-1}}_{n_i}$ replaced by $\widetilde A_k$, $\widetilde A_\ell$, and $\Psi^W_{n_i}$, respectively. This conjecture seeks to classify homomorphisms from the enveloping algebras of these infinite-dimensional Lie algebras to Weyl algebras through the explicitly constructed homomorphisms.