You'd be right if the high prices were caused by increased demand. But not if the high prices are caused by lower supply. In the case of low supply, the cost of finding and catching the rare fish should offset the profit to be had by selling them.
That's how it should happen anyway. As others on this thread have pointed out, there are pecularities with this resource that make it harder to predict.
Let's assume that that's the case. While an equilibrium may eventually be achieved that prevents the fish from being hunted to extinction, it will still stabilize at fish population that is very much smaller than it is today, with no prospect of it coming back.
I think we can do better. In fact, I believe that the ocean fisheries can one day be more abundant than they were in the wild state.