The low-degree automorphic quotient conjecture

Let $\pi_1$ and $\pi_2$ be non-isomorphic unitary cuspidal automorphic representations of $\operatorname{GL}_{d_1}(\mathbb{A}_{\mathbb{Q}})$ and $\operatorname{GL}_{d_2}(\mathbb{A}_{\mathbb{Q}})$, respectively, and let $\Lambda(s,\pi_i)$ denote their completed $L$-functions. Low-degree automorphic quotient conjecture. If $d_2-d_1\leq2$, then

$$\frac{\Lambda(s,\pi_2)}{\Lambda(s,\pi_1)}$$

has infinitely many poles. This is proposed as a consequence of a converse-theorem picture for degree-$2$ Dirichlet series, which would classify suitable objects by modular, Maass, or quadratic-field Hecke $L$-functions. The conjecture is presented as an expected statement and no resolution is given.