No it wont. BCC is secretly supported by the Chinese mining megacorp. That means whenever this cryptocurrency's price falls, there are interest groups out there who will give it its support. Core knows this and they cant do anything about it except call it names on Twitter.
I don't understand this. If by "support" you mean they will buy $100+ million worth of BCH to make its price rise again, for how long they can do that? Won't they be losing their money, big money, providing that support? I mean even if you have huge amounts of money, you lose couple of billions and stop, right? And then the price goes down.
Things are more intricate than that
They don't have to support Bitcoin Cash indefinitely since people will run out of their BCH tokens sooner or later even if all of them choose to get rid of these tokens. Obviously, quite a few BCH holders aren't going to sell these coins, and some may even be looking to buy more. Anyway, after this initial stage of rampant sell-offs and buy-ups is over, their hands will be untied, and they can proceed to the next level of dethroning Bitcoin (as they likely think). You also seem to discard the possibility that the cartel might be selling regular bitcoins for BCH to prop up the price of the latter
I may be wrong but I see it like this. If you are selling something you have to make the price competitive. In other words the price should be lower than in other places, otherwise people will be buying in those places and not from you. In our case if we are selling Bitcoins for Bitcoin Cash we should make people pay less Bitcoin Cash for BTC than in other places to attract buyers. Yes, such act would prop up the price of BCH, but at what cost. We'll be losing money. In fact selling Bitcoins for lower amount of Bitcoin Cash is the same thing as buying Bitcoin Cash for higher amount of Bitcoins
This is certainly not the only possibility
You can sell original bitcoins for dollars and get market rates. And then use dollars to bid up the price slowly. But this is ultimately irrelevant since you may be selling for a loss today to earn decent profits tomorrow. In fact, when you buy some asset you hope that it will rise in the future even if its price continues to decline after you make a purchase. But this is obviously not all. The cartel may not be looking for short term profits (like in "buy low sell high"). They are ultimately looking for control, and they don't care about how much it could cost them in dollar or Bitcoin terms since if they control Bitcoin (under whatever name), they control the whole market (after all, they could break even entirely via mining fees in the end)
Well, if they are people with dozens of billions USD and their aim is to control the whole market, they may try, good luck with that, but I think they will fail. Fortunately lately, I mean 15 years or so, we are living in the world where, thanks to the internet, what is really the best wins. No matter how much money you spend on promotion, if there's something which is better in the same field with the product you promote, you will fail.