Nakai's conjecture on higher derivations and regularity

Let $k$ be a field of characteristic zero and let $A$ be a $k$-algebra of finite type. For each integer $q\ge 1$, let $Der^q(A)$ denote the module of differential operators of order at most $q$ and let $der^q(A)$ be the $A$-submodule generated by compositions of at most $q$ elements of $Der^1(A)$. Nakai's conjecture. If

$$der^q(A)=Der^q(A)$$

for each integer $q\ge 1$, then $A$ is regular. Grothendieck proved the converse implication for finite-type regular algebras. The conjecture was stated by Mount and Villamayor in 1973 and is known for affine rings of irreducible algebraic curves; it remains open in general. It also implies the Zariski–Lipman conjecture.