Mathieu-type admissible-function conjecture for $G_2$

Let $S^*$ denote the unit circle, and let $\tilde J_{G_2}$ be the weight function associated with the paper's Euler-angle parametrization of $G_2$. A function

$$f:[0,1]^6\times(S^*)^8\to\mathbb{C}$$

is $\frac{1}{4}$-admissible if it has the corresponding finite Laurent-type expansion with exponents in $\bigcup_{j=1}^4\frac{1}{j}\mathbb{Z}$ and coefficients polynomial in the real variables and their square-root factors. Let $\operatorname{Sp}(f)$ denote its spectrum. The $G_2$ admissible-function conjecture. If

$$\int_{(S^*)^8}\int_{[0,1]^5}\int_0^{S(x_5)}f^P\tilde J_{G_2}=0$$

for every $P\in\mathbb{N}$, then the zero vector does not lie in the convex hull of $\operatorname{Sp}(f)$. This is the auxiliary conjecture assumed in the paper to prove Mathieu's conjecture for $G_2$; the paper does not provide a resolution.