The weighted Laurent-polynomial Mathieu conjecture

Let $M,N$ be non-negative integers. Let $f,g\in\mathbb{C}[x_1,\ldots,x_N,z_1^{\pm1},\ldots,z_M^{\pm1}]$, and let $\delta\in\mathbb{C}[x_1,\ldots,x_N]$ be a monomial of odd degree in all variables. Let $\int_{(N,M)}$ denote the integral used in the paper.

Weighted Laurent-polynomial Mathieu conjecture. If

$$\int_{(N,M)}f^n\delta=0$$

for all $n\in\mathbb{N}$, then

$$\int_{(N,M)}f^n g\delta=0$$

for all but finitely many $n\in\mathbb{N}$.

The conjecture is formulated to imply Mathieu's conjecture while eliminating reference to Lie groups. The case $N=0$ is known from the supplied context, while the general case remains open; only partial results are mentioned for $N=1$, $M=0$.