I think fiat over lightning is a really bad idea. I'm surprised more people aren't talking about it. It has so many perverse incentives, its not even funny. I could see it as a convenience layer for a merchant perhaps, but I don't think this fantasy of a bunch of HODLers running a network of exchange nodes is going to go anywhere.

Here's a potential application for it:
a:

• Customer pays merchant in bitcoin with lightning
• Merchant converts bitcoin to dollars for his own internal accounting

b:

• Customer pays merchant in dollars with lightning
• Merchant keeps payment in dollars

When merchant ends up with too much bitcoin and needs more dollars:

• Merchant opens bitcoin channel with liquidity exchange
• Liquidity exchange opens dollar channel with Merchant
• Merchant exchanges bitcoin for dollars with lightning with circular route

Here's the problem with this all; How do the fiat channels settle when they are closed if they aren't using a stablecoin? Anytime you open a fiat channel, you'll need to have a written contract with your channel partner to settle manually through your bank.