The multiple-Dirichlet-series pole conjecture for unitary moments

Let $A_{M,N}(s_1,\dots,s_k;z_1,\dots,z_k;w)$ be the multiple Dirichlet series defined in the source, and set

$$\sigma_{J,H}(w):=w+\sum_{j\in J}s_j+\sum_{h\in H}z_h-\frac{|J|}{2}-\frac{|H|}{2}.$$

Unitary multiple-Dirichlet-series pole conjecture. The function

$$\left(\prod_{\substack{J,H\subseteq\{1,\dots,k\}\\|J|=|H|}}(\sigma_{J,H}(w)-2)\right)\left(A_{M,N}(s_1,\dots,s_k;z_1,\dots,z_k;w)-\prod_{j=1}^k\zeta(s_j)\zeta(z_j)\right)$$

has a holomorphic continuation to a tube domain containing $(s_1,\dots,s_k,z_1,\dots,z_k,w)=(\frac12,\dots,\frac12,2)$ and is polynomially bounded in vertical strips there. The conjecture asserts that, after removing the $q=1$ zeta-factor contribution, only the indicated shifted pole hyperplanes occur near the central point.