The admissible-function convex-hull conjecture for $SU(N)$

Let $N\geq 2$, set $k=\frac{N(N-1)}{2}$ and $l=\frac{N(N+1)}{2}-1$, and let $S^*$ denote the relevant multiplicative parameter space. A $SU(N)$-admissible function is a function

$$f:[0,1]^k\times(S^*)^l\rightarrow{\mathbb{C}}$$

with finite expansion $f(x,z)=\sum_{\vec m}c_{\vec m}(x)z^{\vec m}$, where the exponents have components in $\bigcup_{j=1}^N\frac1j\mathbb{Z}$ and the coefficients are complex polynomials in the variables $x_i$ and $\sqrt{1-x_i^2}$. Let $\mathrm{Sp}(f)$ be the set of exponent multi-indices with nonzero coefficient. $SU(N)$ admissible-function conjecture. If

$$\int_{[0,1]^k}\int_{(S^*)^l}f^P J_{SU(N)}=0$$

for every $P\in{\mathbb{N}}$, then $\vec 0$ does not lie in the convex hull of $\mathrm{Sp}(f)$. This is proposed as a complex-analytic reduction of Mathieu's conjecture for $SU(N)$; its status is not resolved by the supplied text.