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!)