As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as the properties thereof are. So, letting $n$ and $n'$ be two distinct positive integers, $\pi$ (respectively $\pi'$) an automorphic representation of $\operatorname{GL}\_{n}(\mathbb{A}\_{\mathbb{Q}})$ (respectively of $\operatorname{GL}\_{n'}(\mathbb{A}\_{\mathbb{Q}})$), is it true that the Rankin-Selberg convolution $\pi\times\pi'$ of $\pi$ and $\pi'$ is an automorphic representation of $\operatorname{GL}\_{n.n'}(\mathbb{A}\_{\mathbb{Q}})$?


Thanks in advance.


Edit October 3rd, 2021: as of today, how has the situation evolved?



Rankin-Selberg convolution and product of degrees
I first disclaim being up-to-date on the precise issue in the question! Given that:


The only truly interesting example I know to have been definitely worked out, that exactly fits the question is Ramakrishnan's result from 2000 (Annals) which proves that the Rankin-Selberg convolutions for $GL\_2\times GL\_2$ (when there's no pole!) are standard cuspidal $L$-functions for $GL\_4$, as would be expected from Langlands Functoriality. (The case that there's a pole is the case that the convolution is of a cuspform/cuspidal-repn and its contragredient, so that the convolution factors as the symmetric square and a zeta.)


The rather boring case of $GL\_1\times GL\_2$ has been known for a long time...


The Cogdell-Jacquet-PiatetskiShapiro approach to modularity by converse theorems already required Ramakrishnan to treat some quite delicate aspects of the triple convolution on $GL\_2$, that is, $(GL\_2\times GL\_2)\times GL\_2$, since to prove modularity on $GL\_n$ we need to know good behavior of convolutions $... \times GL\_m$ with $m\le n-2$ (although it is conjectured, I think, that less would suffice).


Thus, the question amounts to asking about good behavior of triple convolutions $GL\_m\times GL\_n \times GL\_q$ with $q$ up to $mn-2$ (with the current state of knowledge, or at least mine at last notice, about converse theorems).


So, even though Henry Kim and collaborators have exhaustively examined/exploited the Langlands-Shahidi method to see which such $L$-functions are obtained (via exceptional groups, etc), already to show that a convolution on $GL\_2\times GL\_3$ (without poles) came from $GL\_6$, one would need $GL\_2\times GL\_3\times GL\_q$ with $q=1,2,3,4$. I very vaguely remember seeing something (Henry Kim?) about $GL\_2\times GL\_3\times GL\_3$ or maybe $GL\_2\times GL\_3\times GL\_4$ or so, but I've not been to re-locate it.


In any case, there is a very finite supply of Langlands-Shahidi and/or Rankin-Selberg integral repns of the relevant triples known, with no real prospect of improvement in the near future. E.g, $GL\_2\times GL\_4\times GL\_q$ with $q=1,2,3,4,5,6$ are surely not all known, so modularity for $GL\_2\times GL\_4$ would not be proven by current converse theorems.


(If someone can find those few triple cases, it would be helpful!)


