This should not come as much of a surprise as the lower fuel tank is mounted to the chassis in four places with rubber isolation grommets, which make for poor electrical conductivity. The Green conductor is essential to the fuel pump circuit, as it provides the required connection to chassis ground.
Perhaps a more useful test would be to check for voltage (+12vdc) between the Brown conductor and chassis ground (as well as the Green conductor) the next time the bike quits running. If voltage is not present on the Brown conductor and chassis ground, then the Fuel-Cut Off Relay (or some component of the power source circuit) is suspect. On the other hand, if voltage is present on the Brown conductor and chassis ground, but not present with the Green conductor, then the Green conductor (or some component of the chassis ground circuit) is suspect.
The Fuel-Cut Off Relay inspection should include an examination of the condition of the connectors inside the socket that the relay connects to, a few members have noted discoloration/corrosion in this area.
Of course if the voltage checks out, and +12vdc is present between the Brown conductor and the Green conductor, then the Fuel Pump (or some component of this circuit inside the fuel tank) is suspect.
One final thought, when the bike quits running, check for voltage (+12vdc) between the Brown conductor, and chassis ground whilst the conductors are still connected to the Fuel Pump, if there is low or no voltage present, check for voltage again, but this time with the Brown conductor disconnected. If voltage is present only with the Brown conductor disconnected, that would be an indication that the fuel pump circuit cannot pass sufficient current (amps). The usual suspects in this situation are dirty, or corroded contacts, and connections.