The non-automorphy conjecture for truncated multiple Dirichlet series

Let $A$ be the integer matrix defining the multiple Dirichlet series $D_{A,\boldsymbol{\Pi}}(\boldsymbol{s};\boldsymbol{\omega},\boldsymbol{\omega'})$, let $\boldsymbol{\Pi}=(\Pi_1,\dots,\Pi_t)$ be the tuple of automorphic representations, and let $\boldsymbol{\omega},\boldsymbol{\omega'}$ be the twisting parameters. Assume $s_1=\dots=s_t=s>1$. Say that a matrix is equivalent to $A$ up to the Gaussian-elimination row operations described in the paper. Non-automorphy conjecture. If every matrix equivalent to $A$ necessarily contains a row with three nonzero entries, and at least one $\Pi_j$ has a cuspidal component of rank at least $2$, then

$$D_{A,\boldsymbol{\Pi}}(\boldsymbol{s};\boldsymbol{\omega},\boldsymbol{\omega'})$$

is not a product of shifts of automorphic $L$-functions. The conjecture formalizes the expectation that truncating Fourier coefficients in this setting destroys automorphic structure; no resolution is stated.