The pursuit of the block reward pits miners against one another in a race for the reward. In the presence of a desirable block reward, miners who are competing against one another will always push the blockchain forward. However, people have confused what this means. This notion that adversarial competition strengthens security only applies to the pursuit of the next block reward; it does not apply to anything else in the Bitcoin ecosystem. Once the block reward falls below other possible incentives for miners, there could be a massive phase shift in security.

We have no idea what happens then.