The Laplace-transform formulation of the Image Conjecture

Let ${\mathcal M}$ be the subspace of polynomials $f(\xi,z)\in{\mathbb C}[\xi,z]$ such that ${\mathcal Z}(f)$ has no holomorphic part, equivalently such that the Laplace transformation ${\mathcal L}(f)(\xi)$ has no $\xi^{[-1]}{\mathbb C}[\xi^{-1}]$-part. The Laplace-transform formulation of the Image Conjecture. For any $f(z),g(z)\in{\mathbb C}[\xi,z]$ with $f^m\in{\mathcal M}$ for every $m\geq1$, one has $f^m g\in{\mathcal M}$ for all sufficiently large $m$.

This is presented as a restatement of the image conjecture using the preceding description of $\operatorname{Im}\Theta$ and the multidimensional Laplace transformation. It is therefore not a new independent conjecture, but a reformulation of the preceding image conjecture.