Both leds are connected to ground.

I added the PAUSE as you mentionned. Now, Led0 is blinking slowly without any button pressed.

Before I added the PAUSE, the Led0 was always ON until I pressed the button to make Led1 switch ON.

So, unfortunately, no big change.

I can see I never get a Z state (between Low and High TTL).

I also measured my resistors and the voltage levels are correct.