Accordingly, "miner revenue from transaction fees with active use of LN is lower than today’s levels until a threshold of 20 million LN users, each transacting 10 times/day, is breached."[1]
[1] https://medium.com/@matteoleibowitz/bitcoin-disinflating-to-death-b4ba7b691969
Medium articles by some no-name guys aren't the best source of information.
LN has limitations to the size of transactions, and not everyone would want to put every single one of their transactions on LN, so there will always be demand for on-chain transactions. Even now we have huge congestions from time to time, and we aren't even close to mass adoption. So, I wouldn't worry that there will be not enough transactions on chain to create profitable enough fees for miners.