The XZ conjecture for $G_2$

Let $f:[0,1]^6\times\mathbb{T}^8\to\mathbb{C}$ be an admissible function in the sense of the preceding definition, with spectrum $\operatorname{Sp}(f)$. Let $S(\xi_1)$ denote the upper integration boundary and define

$$\tilde{J}_{G_2}(x_1,\ldots,x_4,\xi_1,\xi_2):=\xi_1\xi_2\left[\xi_1^2\left(16(1-\xi_2^2)^3+9(1-\xi_2^2)-24(1-\xi_2^2)^2\right)-(1-\xi_1^2)(3\xi_2-4\xi_2^2)^2\right]\left[\xi_1^2(1-\xi_2^2)-(1-\xi_1^2)\xi_2^2\right]x_1x_2x_3x_4.$$

The XZ conjecture for $G_2$. If

$$\int_{\mathbb{T}^8}\int_{[0,1]^5}\int_0^{S(\xi_1)}f^P\tilde{J}_{G_2}\,d\xi_2\,d\xi_1\,dx_1\cdots dx_4\frac{dz_1}{z_1}\cdots\frac{dz_8}{z_8}=0$$

for all $P\in\mathbb{N}$, then $\vec{0}$ does not lie in the convex hull of $\operatorname{Sp}(f)$. The paper presents this as the remaining implication needed to obtain the Mathieu conjecture for $G_2$.