If this was the whole board, white would indeed be dead. This is why it’s so important who takes the ko first! (in the orginal problem, there are other ways to start the ko which lead to white taking first, but black of course chooses the one which is worst for white).
But all hope is not lost for white, if there is some big ko threat elsewhere on the board:
If white plays A here, there are two possible results:
- Black responds at B, letting white capture back at S19.
- Black captures at Q18, letting white kill with B.
This is the nature of all ko’s, it will be captured back and forth as long as both sides can find big enough threats. Eventually, one side gets to finish the ko, and the other gets some compensation elsewhere.