I don’t see how that can happen if it’s built right. There’s a diode between the buffer op amp and the output (and resistor to LED). If the capacitor voltage is negative then current cannot flow backwards through the diode, so there can be no voltage drop across the load to ground, so the output must be zero volts.
Furthermore the capacitor shouldn’t be charging quickly up to a negative value. The incoming trigger should charge it quickly to a positive value, and then it’ll decay to about -10 V, but it shouldn’t charge quickly up to that.
So I’d say something’s wrong with your build. See:
Since you have a scope, check directly:
- Voltage on the capacitor, should shoot up to (about) +10 V and then decay to -10 V
- Voltage on the op amp pin 1, should alternate between (about) +10 V and -10 V
You did plug it in, right?