Zhao's locally nilpotent derivation conjecture for theta-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

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

A derivation or $\mathcal{E}$-derivation is locally nilpotent if, for every $a\in\mathcal{A}$, some positive iterate sends $a$ to zero. The notions of a $\vartheta$-ideal and a $\vartheta$-Mathieu-Zhao space are those used in the source.

LNED conjecture. Let $K$ be a field of characteristic zero, let $\mathcal{A}$ be a $K$-algebra, and let $\delta$ be a locally nilpotent derivation or $\mathcal{E}$-derivation of $\mathcal{A}$. Then for every $\vartheta$-ideal $I$ of $\mathcal{A}$, the image $\delta(I)$ of $I$ under $\delta$ is a $\vartheta$-Mathieu-Zhao space of $\mathcal{A}$.

This is the second conjecture in the pair attributed in the source to Wenhua Zhao. The supplied context does not define $\vartheta$-ideals or $\vartheta$-Mathieu-Zhao spaces, and gives no resolution status.