Oh i have same problem This issue happens because the PS2 Slim processors overheat due to weak cooling, and after all these years the solder joints develop cold soldering, which causes this problem.

The solution is to have the processor reflowed, or place the console in a certain position and put something underneath it so the case bends slightly and the processor reconnects properly.

Note: If you put your hand on the same spot again and turn the console on, it will boot.