I feel like a broken record saying this: Demand for the currency will fluctuate. If the supply is fixed, the price will be demand-driven. Which means volatile. Fiat currency has a central bank with active control over the supply, so it is capable of keeping a stable price if the central bank does its job.
http://www.bzarg.com/p/what-bitcoin-shows-us-about-how-money...
It wouldn't be too hard to make a inflationary coin, just mint coins proportional to the estimated GH of the network. That would give the coin a price ceiling.
I am really looking forward to LN being really tested, but the thing that worries me is that the security model relies on txing before a certain block. If we see a large amount of the traffic move from on chain to LN then we could get ourselves into a situation where there is a run on the chain to close channels. If there is a large market maker that attempts to steal from a channel that is being used to route many others we could see a mass closing of channels which could push the fees high enough to make it cost prohibitive.
LN isn't a panacea it actually changes the security model.
http://www.blunderingcode.com/a-lightning-network-in-two-pag...
For instance I would be happy to buy a cryptokitty or two at the ~1 USD price range, but if the tx fees are ~1 USD then it's obviously a no go.
But if the valuations are anywhere close to proper we need to have significantly higher throughput.
If I can't send someone 10 bucks on bitcoin it loses a lot of value for me. I've previously used it to pay for lunch/poker w/e when I don't have my wallet and someone that I am with is interested in it. Even litecoin is hitting 1 USD avg tx fee.