The radical of the image of a triangular E-derivation

Let $K$ be the algebraically closed field of characteristic zero fixed in the paper, and let $\delta=I-\phi$ be an E-derivation of $K[x_1,x_2,x_3]$, where, for $i=1,2$, $\phi(x_i)=\lambda x_i+x_{i+1}$ and $\phi(x_3)=\lambda x_3$ for some $\lambda\in K$. Let $\mathfrak{r}(M)=\{a\in K[x_1,x_2,x_3]:a^m\in M\text{ for all sufficiently large }m\}$ denote the radical of a Mathieu-Zhao space. Radical-image generation claim. The radical $\mathfrak{r}(\operatorname{Im}\delta)$ is the $K$-vector space generated by the monomials

$$x_1^{i_1}x_2^{i_2}x_3^{i_3}$$

for all $i_3\geq i_1+1$ and $i_1,i_2,i_3\in\mathbb{N}$. The supplied excerpt presents this as a result in the paper, but gives no explicit resolution status beyond the statement itself.