So, I thought I would try it. I don't have a 2N3819 but I do have a J112 N-channel JFET. The circuit is shown below:
Power is supplied by a bench power supply set to 3V current limited to 10mA. The "antenna" is a 5-inch long piece of wire.
So how does it work? A JFET has extremely high input impedance - hardly any current flows into the gate - it is voltage controlled. Current flow from drain to source is determined by the voltage differential between the gate and the source. The drain to source current flow is highest when the gate to source voltage differential is zero. As the gate to source voltage differential becomes more negative the drain to source current flow decreases. At the pinch-off (sometimes called cutoff) voltage the drain to source current essentially ceases. In the detector circuit, the gate/antenna becomes negative relative to the source and the FET turns the LED off gradually as the statically charged object comes closer. I demonstrate it in the following 40-second video: