The Factorial Conjecture in dimension m

Let $m\in\mathbb N_+$, let ${\mathbb C}^{[m]}={\mathbb C}[X_1,\ldots,X_m]$, and let ${\cal L}:{\mathbb C}^{[m]}\to{\mathbb C}$ be the linear map defined by

$${\cal L}(X_1^{l_1}\cdots X_m^{l_m})=l_1!\cdots l_m!.$$

Factorial Conjecture. For all $f\in{\mathbb C}^{[m]}$,

$$(\forall k\in\mathbb N_+)\,{\cal L}(f^k)=0\Rightarrow f=0.$$

This is the factorial conjecture recalled from the earlier literature; its general validity is the motivating problem for the stronger formulation studied in the paper.