The non-Mathieu–Zhao image conjecture for simple derivations

Let $K$ be a field, let $n\geq 2$, and let $D$ be a simple derivation of the polynomial ring $K[x_1,\ldots,x_n]$. A $K$-subspace $M$ of a ring is a Mathieu-Zhao space if, whenever $a^m\in M$ for all $m\geq 1$, one has $ba^m\in M$ for every $b$ and all sufficiently large $m$. The image of $D$ is denoted by $\operatorname{Im}D$. The non-Mathieu-Zhao image conjecture. The subspace $\operatorname{Im}D$ is not a Mathieu-Zhao space. The paper proves this for images of simple Shamsuddin derivations and for some simple derivations in dimension three, while the assertion for all simple derivations in dimensions $n\geq 2$ remains open.