The weighted Laurent-polynomial spectrum conjecture

Keep the notation above: $M,N$ are non-negative integers, $f,g\in\mathbb{C}[x_1,\ldots,x_N,z_1^{\pm1},\ldots,z_M^{\pm1}]$, $\delta\in\mathbb{C}[x_1,\ldots,x_N]$ is a monomial of odd degree in all variables, and

$$f=\sum_{\mathbf m\in\mathbb{Z}^M}c_{\mathbf m}z^{\mathbf m},$$

where $c_{\mathbf m}\in\mathbb{C}[x_1,\ldots,x_N]$. Define the spectrum

$$\operatorname{Sp}(f)=\{\mathbf m\in\mathbb{Z}^M\mid c_{\mathbf m}\ne0\}.$$

Weighted Laurent-polynomial spectrum conjecture. If

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

for all $n\in\mathbb{N}$, then $\mathbf 0$ is not in the convex hull of $\operatorname{Sp}(f)\subset\mathbb{R}^M$.

This is presented as a generalization of the $xz$-conjecture and would give a spectral criterion relevant to the weighted Laurent-polynomial Mathieu conjecture. The supplied text gives no resolution beyond the surrounding partial results, so the conjecture remains open.