The thing that could happen is that it will maybe get stable for a while but I do not think this is gonna happen honestly.
If the price rises a lot of people will sell it again and others will buy without any doubts.
As quoted price won't get stable.
The price of bitcoin starts to decrease gradually, if none spends it. The major reason is that the circulation of bitcoin decreases which needs to be in a increased range or in a stable manner. But such scenarios of continuous decrease due to such situations won't happen.
Methinks, there should be some temporal lag before the price starts to decline in earnest. At first, it might even be rising for some time because no one spending bitcoins would likely mean the supply side contraction. In other words, the demand for bitcoins would initially surpass the supply of them thanks to no one selling bitcoins. But ultimately it would collapse too since buyers would soon lose interest in bitcoins as well.