Around two years ago, one could find on the Internet a lot of rumors that CBS was unhappy with Stephen Colbert's ratings and were fixing to either swap his time slot with James Corden's or dump him entirely. I never thought there was anything to those rumors other than that some people who didn't like his show just liked reading that it was doomed. But Colbert's still there and he's doing great. In fact, as this article notes, greater than anyone else on in that day part.
I like his show a lot, though I might like it a little more if he talked less about Trump and more about other things. But what he's doing is working fine so I don't expect a change.