Doubtful, Litecoin has been keeping a pretty steady ratio to Bitcoin, if anything I expect it to decrease with the influx of miners running from the ASICs. That's why I am mining ATM, not buying.
Wouldn't the increase in popularity cause an increase in price?
Not necessarily, it would require a lot more dev support. Bitcoin spent quite a while at a small dollar value. The spike in value really came when the speculators arrived. I have no doubt cryptocurrency is the future, but I think Litecoin has a bit of growing up to do first. If it hits $10 all the speculators will bail and possibly take the coin down with them.