The image conjecture for commuting differential operators

Let $t\in\mathbb C$, and let

$$\operatorname{Im}(\xi-t\partial):=\sum_{i=1}^n(\xi_i-t\partial_i)\mathbb C[\xi,z]$$

be the image of the commuting differential operators $(\xi_i-t\partial_i)$. A subspace $\mathcal M$ of a commutative algebra $\mathcal A$ is a Mathieu subspace if, whenever $a^m\in\mathcal M$ for every $m\geq1$, then $ba^m\in\mathcal M$ for every $b\in\mathcal A$ and all sufficiently large $m$. The image conjecture. For every $t\in\mathbb C$, $\operatorname{Im}(\xi-t\partial)$ is a Mathieu subspace of the polynomial algebra $\mathcal A[\xi,z]$. This conjecture implies the Jacobian conjecture and is equivalent to certain special cases of it; its general validity remains open.