On the contrary, the higher the price of bitcoin rises, the greater should be its volatility.
I am pretty sure certain about the opposite.
As the price of bitcoin increases, getting closer and closer to his final price given by the hyperbitcoinisation phase, the less impact will be given by the news about bitcoin. So the less impact on price, and then volatility will be greatly reduced.
Also trading activity will be huge, and no single dealer will be able to impact the price significantly.
Think at the BTC/USD as the new EUR/USD.