Just like in the 90's when they tried to turn "Politically Correct" into a conservative expression. It failed very fast. The same will happen to "woke".
There was a time when they did label the right something and it stuck. After spending the 1920's praising everything Hitler, Mussolini and Stalin said, once word started getting out, first about Stalin and later Hitler and Mussolini, the left went to great lengths at pushing Nazism and fascism off on us. They got that stupidity in to colleges and now people just accept it. It's complete and utter crap. Nazis and fascists are socialists and always have been.