Zhao's Image Conjecture for commuting differential operators

Let $\mathbb{C}[z]$ be a polynomial ring and let $\mathbb{D}[z]$ be the set of differential operators $\Phi$ of $\mathbb{C}[z]$ of the form

$$\Phi=h(z)+\sum_{i=1}^n c_i\frac{\partial}{\partial z_i},$$

for some $h(z)\in\mathbb{C}[z]$ and $c_i\in\mathbb{C}$. For a subset $\mathcal C=\{\Phi_i\mid i\in I\}$ indexed by $I\subset\mathbb Z$, define

$$\operatorname{Im}\mathcal C=\sum_{i\in I}\Phi_i\mathbb{C}[z].$$

Call $\mathcal C$ commuting when its elements commute pairwise. Zhao's Image Conjecture. For every commuting subset $\mathcal C\subset\mathbb D[z]$, $\operatorname{Im}\mathcal C$ is a Mathieu–Zhao subspace of $\mathbb C[z]$. The source says that the Jacobian Conjecture is a special case of this conjecture and gives no resolution evidence for the general statement.