Nakai's conjecture for finitely generated algebras

Let $k$ be a field of characteristic $0$, and let $A$ be a finitely generated $k$-algebra. For each $q\geq 0$, let $Der_k^q(A)$ denote the set of $q$-th order derivations of $A$ over $k$.

Nakai's conjecture. If $Der_k^n(A)$ is generated by $Der_k^1(A)$ for every $n\geq 2$, then $A$ is regular.

The conjecture asks whether differential operators generated by derivations can detect singularities. It is known in several special cases, including irreducible curves, monomial ideals, and certain hypersurfaces, while the general case remains open; the supplied parser gives no resolution status.