Almost 5 years ago (time flies), I asked in [Rankin-Selberg convolution and product of degrees](https://mathoverflow.net/questions/194770/rankin-selberg-convolution-and-product-of-degrees?r=SearchResults) whether the Rankin-Selberg convolution of two automorphic representations of respectively $\operatorname{GL}\_{n}(\mathbb{A}\_{\mathbb{Q}})$ and  

$\operatorname{GL}\_{n'}(\mathbb{A}\_{\mathbb{Q}})$ gave rise to an automorphic representation of $\operatorname{GL}\_{n.n'}(\mathbb{A}\_{\mathbb{Q}})$. Paul Garrett answered it by giving the known cases where this was proven at that time.


Have there been breakthroughs so far getting us any closer to such a general result?


Edit August 12th 2022: what about now? I would be especially interested in the case where the degree of the Rankin-Selberg convolution L-function is a prime power, that is $d\_{F^{\otimes k}}=(d\_{F})^{k}$ with $d\_{F}\in\mathbb{P}$. A proof of this equality would entail that the following weakening of Goldbach's conjecture: "every composite even integer is the sum of 2 prime powers" implies that an L-function of composite even degree is a twist of a non primitive L-function.



Rankin-Selberg convolution and product of degrees as of Christmas 2019
[Newton and Thorne](https://arxiv.org/abs/1912.11261) proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}\_2(\mathbb{A}\_{\mathbb{Q}})$ corresponding with a holomorphic cuspidal newform of even integral weight $k\geq 2$, squarefree level, and trivial central character, then for each $n\geq 1$, the $n$-th symmetric power lift $\mathrm{Sym}^n \pi$ is a cuspidal automorphic representation of $\mathrm{GL}\_{n+1}(\mathbb{A}\_{\mathbb{Q}})$. We have the standard identity


$\mathrm{Sym}^n \pi\otimes\mathrm{Sym}^n\pi = \boxplus\_{j=0}^n \mathrm{Sym}^{2j}\pi$,


so by Newton--Thorne, this is an isobaric automorphic representation of $\mathrm{GL}\_{(n+1)^2}(\mathbb{A}\_{\mathbb{Q}})$. This is not yet known to hold for $\pi$ corresponding with Hecke--Maass forms. Thus this is a thin set of examples, but I find it noteworthy nonetheless. This showed up on the arxiv the day before you posted your question.


