I've examined my 650D harness a couple of times (I have a box of mystery electronics to hook up), and my light checker connector looks just like yours, and is in the same location. The problem with guessing that this is the light checker connector is that all the diagrams show a yellow, a green/white and a black wire in the connector. Yours appears to have a green/yellow in place of the yellow). This doesn't match up with any wiring diagram I've seen.
Another problem with the light checker guess is that only the 650A through D diagrams show a 3-wire connector. The E & Fs shows 7 wires connecting to the light checker.
Still, I'm thinking the same as weekendrider, that it might go to the light checker. Maybe a mid-model-year re-design. One clue that it may indeed be to the light checker is that the yellow wire in the wiring diagrams becomes a green/yellow wire on its way to the front brake light switch. It may be that this splice was done somewhere else in your harness when manufactured, and therefore doesn't agree with the wiring diagram.
One way to test whether it is tied to the lights is to test to see if green/yellow wire has continuity with the non-powered side of the brake light switch (should be a yellow wire with a plug-connector end. Or, if the battery is connected, the green/yellow wire would show power when either the front or rear brake switches are activated.
Of course, it could just be a GNDN plug (Goes Nowhere, Does Nothing). Do you have an electrical component that plugs into it? If so, picture?
The following is just grain-of-salt stuff. If the above is wrong, just consider this more useless information:
If the connector in the picture is to the light checker, the way these usually work is that the checker would test for continuity through both the tail and brake light filaments (by allowing a very small amount of current to pass through the filaments and a resistor to ground, with the key on. If power is present, the checker does nothing. If absent, the checker acts as a flasher by intermittently connecting the Brake Light Indicator on the dash (via the green/white wire) to ground, completing the circuit for the indicator and causing it to flash. Although I'm not sure, I would guess that the light checker would also test for power from the other wire (In the schematics, a yellow wire, but in your case a green/yellow?) when either brake light switch is activated. If it doesn't see power to this wire, then the checker would do the same and cause the Brake Light Indicator to flash.