YES THEY WILL
Because behind all this cryptocurrency is technology: the Blockchain.
And BTC made good advertisment for this tech, so the public is now aware and wont dump it.
But not like other cryptos, BTC is just blockchain, nothing more and if we are realistic BTC is at the moment the oldest, most expensive and got the fewest real world value.
Also a friend of mine made a bachelor-thesis about cryptocurrency and found out at university: there is no correlation between cryptos.
Cyptocurrencies/Blockchain is a young market, its evolving, growing and propertly will survive.