I'm not sure if you have a different converter than I do, (mine is a WFCO WF-8900) but mine goes into float mode (13.2 volts) when the battery is fully charged. It stays there until something is turned on ( light, pump, etc.). Then it goes back into charge mode. It never turns off.
Though I'm not familiar with every converter out there, I'm thinking that all converters work the same way.