The idempotent conjecture for images of LF derivations

Let $K$ be a field of characteristic zero and let ${\mathcal A}$ be a $K$-algebra, not necessarily unital or commutative. Let $\delta$ be a locally finite $K$-derivation or a locally finite $K$-${\mathcal E}$-derivation of ${\mathcal A}$, and write $\operatorname{Im}\delta=\delta({\mathcal A})$. An element $e\in {\mathcal A}$ is idempotent if $e^2=e$; let $(e)$ denote the principal two-sided ideal generated by $e$.

Idempotent conjecture. If $e\in\operatorname{Im}\delta$ is an idempotent, then

$$(e)\subseteq\operatorname{Im}\delta.$$

This is presented as a weaker version of the conjecture that these images are Mathieu subspaces. It concerns whether the image contains the entire two-sided ideal generated by each idempotent that it contains, and its resolution status is not specified in the supplied text.