If BTC is not dead by then, it will likely have some sort of active user base, meaning non-zero fees.
If the fees are low, verifiers will stop verifying.
If enough stop verifying, the prize pool grows (more network congestion leads to higher fees). This eventually stabilizes at some equilibrium (likely leading to a weaker chain, if profit is the only motivator).
It's also worth noting that if BTC has seen enough adoption to be sustainable for the next 100 years, there are likely many many stakeholders who could/would mine for "altruistic" reasons. E.g. financial institutions and other large holders have incentives to keep the network stable.
Hell, maybe by then they get enough people on board for a BTC 2 that's some weird Ethereum competitor. It's really hard to theorize on that long of a time horizon.