The multiple-Dirichlet-series pole conjecture for elliptic-curve twists

Let $A_{E,M}(s_1,\dots,s_k;w)$ be the multiple Dirichlet series associated with quadratic twists of the elliptic curve $E$, and for $J\subseteq\{1,\dots,k\}$ set

$$\sigma_J(w):=w+2\sum_{j\in J}s_j-|J|.$$

Elliptic-curve multiple-Dirichlet-series pole conjecture. The function

$$\left(\prod_{J\subseteq\{1,\dots,k\}}(\sigma_J(w)-1)\right)A_{E,M}(s_1,\dots,s_k;w)$$

has a holomorphic continuation to a tube domain containing $(s_1,\dots,s_k,w)=(\frac12,\dots,\frac12,1)$ and is polynomially bounded in vertical strips there. This is the conjectural analytic continuation underlying the modified elliptic-curve moment prediction.