I'm curious what your reasoning behind this assumption is
So far we have seen the opposite effect. Bitcoin prices have been rising along with the fees, and the latter seem to have been rising even faster. That can be easily explained by the fact that fees and prices are not very tightly related to each other. Fees mainly depend on the number of transactions and the block size (whether these transactions are genuine or spammy is another question), while the price is determined on exchanges by the balance of supply and demand, while you don't need to make a transaction to make a trade