Weak Nakai 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 $k$. For each integer $q\geq 1$, define $der_k^q(A)$ to be the $A$-submodule of $Der_k^q(A)$ generated by compositions

$$\delta_1\delta_2\cdots\delta_j,$$

where $1\leq j\leq q$ and each $\delta_i\in Der_k^1(A)$.

Weak Nakai conjecture. If $der_k^q(A)=Der_k^q(A)$ for each integer $q\geq 1$, then $A$ is regular.

This weaker form of the Nakai conjecture appeared in previous work and is part of the broader program of detecting regularity through differential operators. The supplied text gives no resolution status for this formulation; the general claim should therefore be treated as open.