Currently, the following $\mathrm{GL}(n)$ converse theorem due to Cogdell and Piatetski-Shapiro is known: For an admissible irreducible representation $\pi$ of $\mathrm{GL}(n, \mathbb{A})$, $\pi$ is automorphic if, for all automorphic representation $\sigma$ of $\mathrm{GL}(m, \mathbb{A})$ with $m\leq n-2$, the Rankin-Selberg $L$-function $L(s, \pi \times \sigma)$ is nice (admits a meromorphic continuation, functional equation, bounded in a vertical strip, ...). Also, it is conjectured that twisting only with $m\leq n/2$ would be enough.
My question is, why one can believe such conjectures? Why half would be enough? I wonder if there's any intuitions or evidences that make us believe the conjecture.