Nope, Jonty, that's not the case in A320. A320 center tank fuel operation logic is the same with Boeing aircrafts (737/757/767/777).
"All fuel pumps (wing pumps & center tank pumps) remain on throughtout the flight. They are fitted with pressure relief sequence valves which ensure that, when all pumps are running, the center tank pumps will deliver fuel preferentially"
FCOM 1.28.10, page 2.
Wing Tanks Pump PushButton - On
"Pump is on but fuel feeds only when center tank pumps delivery pressure drops below threshold"
FCOM 1.28.20 page 1.