Lightning Network will and already has very cheap fees, but this doesn't mean that on-chain fees will stay low too. First, if onchain fees were low, why bother with LN and its channels? Also, LN channels have some limits to their capacity, LN is designed to serve for smaller daily spendings rather than being your saving wallet. So, onchain transactions will still be used for big transfers, and in the future the number of those transfers will keep growing, especially during the price increases, so the fees will keep growing too. In the very long run, the fees will grow huge, easily $500 or $1,000 in the next 20 years, depending on the levels of adoption. But this will be fine, because we'll have Lightning and some other cool technologies.
Fees of 500$ or 1000$? Never.
People will never pay 500$ or 1000$ for that, at this point it will simply not be worth using BTC anymore
During the last hike, a lot of users stopped paying with BTC and just waited, using BTC only when there as no other choice.
If the fees will go to 1000$ (that's half of the median wage in EU) who do you think will keep on paying those?
People, that transfer millions a day? Maybe, but those are not enough to clog the chain.
Just imagine, the network is doing an average of about 400 000tx when the blocks are full. With a 1000$ fee, it would mean 400 million a day in fees or 1.2 billion a month. Even the whales will turn into shrimps at this rate.