Almost 5 years ago (time flies), I asked in 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?