Almost 5 years ago (time flies), I asked in Rankin-Selberg convolution and product of degrees 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 $d_{F}\in\mathbb{P}$.