Zhao's locally finite derivation conjecture for Mathieu-Zhao spaces

Let $K$ be a field of characteristic zero and $\mathcal{A}$ a $K$-algebra. A derivation is a $K$-linear map satisfying the Leibniz rule, and an $\mathcal{E}$-derivation is a $K$-linear map $\delta$ satisfying

$$\delta(ab)=\delta(a)b+a\delta(b)-\delta(a)\delta(b).$$

An endomorphism is locally finite if, for every element, the submodule spanned by its iterates is finitely generated. An $K$-submodule $M$ of $\mathcal{A}$ is a Mathieu-Zhao space if, whenever $a\in\mathfrak{r}(M)$, one has $ba^m\in M$ for all sufficiently large $m$ and every $b\in\mathcal{A}$, where $\mathfrak{r}(M)=\\{a\in\mathcal{A}:a^m\in M\text{ for all sufficiently large }m\\}$.

LFED conjecture. Let $K$ be a field of characteristic zero and $\mathcal{A}$ a $K$-algebra. Then for every locally finite derivation or $\mathcal{E}$-derivation $\delta$ of $\mathcal{A}$, the image

$$\operatorname{Im}\delta:=\delta(\mathcal{A})$$

of $\delta$ is a Mathieu-Zhao space of $\mathcal{A}$.

This is one of two conjectures posed by Wenhua Zhao concerning images of derivations and $\mathcal{E}$-derivations. Its status is not specified in the supplied source.