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!)

conjecturedto be true, by "Langlands functoriality", butprovenin very, very few cases. $\endgroup$ – paul garrett Jan 24 '15 at 17:27