Mathieu-type admissible-function conjecture for $Sp(N)$

Let $N\in\mathbb{N}$, and let $S^*$ denote the unit circle. A function

$$f:[0,1]^{N^2}\times(S^*)^{N(N+1)}\to\mathbb{C}$$

is $\frac{1}{N}$-admissible if it is a finite sum $\sum_{\vec m}c_{\vec m}(x)z^{\vec m}$, where the exponents lie in $\bigcup_{j=1}^N\frac{1}{j}\mathbb{Z}$ and each coefficient is a complex polynomial in the variables $x_i$ and $(1-x_i^2)^{1/2}$. Let $\operatorname{Sp}(f)$ be the set of multi-indices with nonzero coefficient, and let $\tilde J_{Sp(N)}$ be the weight function defined in the paper. The $Sp(N)$ admissible-function conjecture. If

$$\int_{[0,1]^{N(N-1)}}\int_{(S^*)^{N(N+1)}}\int_0^1\int_0^{\xi_N}\cdots\int_0^{\xi_2}f^P\tilde J_{Sp(N)}\,d\xi_1\cdots d\xi_N\frac{dz_1}{z_1}\cdots\frac{dz_{N(N+1)}}{z_{N(N+1)}}dx_1\cdots dx_{N(N-1)}=0$$

for every $P\in\mathbb{N}$, then the zero vector does not lie in the convex hull of $\operatorname{Sp}(f)$. This conjecture is introduced as the condition needed to reduce Mathieu's conjecture to an abelian-type integral for $Sp(N)$; its resolution is not supplied in the paper.