Crusts how nailed it. To prove it, lok at the voltage where the transistor meets the resistor. You should see the same voltage tail there. That indicates there is no "large" current flowing into the LED.
Another thing to prove this effect is to swap the resistor and LED so the resistor becomes a direct read of the current. That voltage will tail off much faster.
Another thing to prove this effect is to swap the resistor and LED so the resistor becomes a direct read of the current. That voltage will tail off much faster.