Seeing as how dreamers are basically the only reason Bitcoin rose to $20k, I see no reason why the trend wouldn't replay itself with Litecoin. Great time to buy IMO.
Well one thing is Litecoin has a much larger supply (3.24x according to Coincap). In terms of market cap & supply as of today a $15k Litecoin is equivalent to ~$50k Bitcoin. The market cap of Litecoin alone would be 0.84 Trillion. It's possible I suppose but I think you'd need some pretty serious institutional buying (or gaming of the system which only lasts so long).
I think that would put a price target of Litecoin at somewhere around $4000-$5000. Not bad considering it's current price today.
Until people actually run nodes to the same distribution of bitcoin, it will always be a reduced security option, and therefore won't command the same price.
The price commanded is whatever the maximum you can get away with selling it to a fool would be, distribution be damned.
The bitcoin price isn't controlled. Supply and demand.