I hadn't really considered this before, but I think there's a good chance they'll do something to increase their take-home if bitcoin becomes cheap enough relative to the mining reward that its security becomes threatened. Right now there are lots of people mining bitcoin at a paper loss due to speculation. If they go out of business after a few years, or are forced to sell because "number goes up" doesn't happen, there will be an exodus of mining power, which could then lead to the Bitcoin network being vulnerable to double spend attacks (these have already happened a number of time on networks with less hash power, like Ethereum Classic [not ethereum])

If double spends looked imminent, or started occuring, I think it would make sense to fork to increase the block reward. Of course, bitcoin has forked before and it becomes a political issue. Personally, the big block wars (and subsequent Bitcoin Cash split) was where I got off the Bitcoin train.

But if a fork happened, and the network that didn't adopt the higher block reward started getting 51% attacked, I do think people would move over to the new network (and that would become "bitcoin")