[Question] How would Logical Decision Theories address the Psychopath Button?

In order to better understand the differences between different decision theories, I have been browsing each and every Newcomblike Problem and keeping track of how each decision theory answers it differently. However, I seem to be coming up short when it comes to answers addressing the Psychopath Button:

Paul is debating whether to press the “kill all psychopaths” button. It would, he thinks, be much better to live in a world with no psychopaths. Unfortunately, Paul is quite confident that only a psychopath would press such a button. Paul very strongly prefers living in a world with psychopaths to dying. Should Paul press the button?

In the FAQ I read, they only gave examples from CDT and EDT, of which CDT says “yes” (because pressing the button isn’t casually linked to whether Paul is already a psychopath) while EDT says “no” (because pressing the button increases the probability that Paul is a psychopath).

So I wonder how Logical Decision Theories (TDT, FDT, and UDT) would address the problem? Unlike Newcomb’s Problem, there is technically only one agent in play, and in the other problem that has only one agent (the Smoking Lesion Problem) the answers of LDT all agreed with CDT. But in this case, CDT doesn’t win.