The functional-equation existence conjecture for non-automorphic Dirichlet series

Let $A$, $\boldsymbol{\Pi}$, $\boldsymbol{\omega}$, and $\boldsymbol{\omega'}$ be parameters for the Dirichlet series $D_{A,\boldsymbol{\Pi}}(\boldsymbol{s};\boldsymbol{\omega},\boldsymbol{\omega'})$. Assume they satisfy the non-automorphy conjecture, namely that the series is not a product of shifts of automorphic $L$-functions under the stated matrix and cuspidality conditions. Functional-equation existence conjecture. There exist such $A$, $\boldsymbol{\Pi}$, $\boldsymbol{\omega}$, and $\boldsymbol{\omega'}$ for which

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

satisfies a functional equation and admits a meromorphic continuation to the entire complex plane. This asserts the possibility of retaining analytic continuation and a functional equation despite the predicted loss of automorphy; no example or resolution is supplied.