The nested-subalgebra classification conjecture for infinite-codimension Witt subalgebras

Let $W_{\geq -1}$ be the Lie algebra of polynomial vector fields on the affine line with degrees at least $-1$, let $\Bbbk$ be the base field, and let $L(f)$ and $L(f,g)$ be the subalgebras associated with polynomials $f,g\in\Bbbk[t]$. Nested-subalgebra classification conjecture. If $\mathfrak{g}$ is an infinite-dimensional subalgebra of $W_{\geq -1}$, then there exist $f,g\in\Bbbk[t]$ such that $f'g\in\Bbbk[f]$ and

$$L(f,g)\subseteq\mathfrak{g}\subseteq L(f).$$

In particular, $\mathfrak{g}$ has finite codimension in $L(f)$. This is stated as a conjectural classification of infinite-dimensional subalgebras, and the paper does not establish it in general.