The XZ conjecture for $SU(N)$

Let $N\geq 2$, and let $f:[0,1]^{\frac{N(N-1)}{2}}\times \mathbb{T}^{\frac{N(N+1)}{2}-1}\to\mathbb{C}$ be an admissible function, meaning that it has a finite expansion in the torus variables with coefficients polynomial in the interval variables and their square roots. Its spectrum $\operatorname{Sp}(f)$ is the set of torus multi-indices occurring with nonzero coefficient. Let $\tilde{J}_{SU(N)}$ be the recursively defined Jacobian factor for the $SU(N)$ decomposition. The XZ conjecture for $SU(N)$. If

$$\int_{[0,1]^{\frac{N(N-1)}{2}}}\int_{\mathbb{T}^{\frac{N(N+1)}{2}-1}}f^P\tilde{J}_{SU(N)}\,\frac{dz_1}{z_1}\cdots\frac{dz_{\frac{N(N+1)}{2}-1}}{z_{\frac{N(N+1)}{2}-1}}dx_1\cdots dx_{\frac{N(N-1)}{2}}=0$$

for all $P\in\mathbb{N}$, then $\vec{0}$ does not lie in the convex hull of $\operatorname{Sp}(f)$. This is the key remaining implication used to deduce the Mathieu conjecture for $SU(N)$; the paper proves the Mathieu conjecture assuming this statement.