If you use the Zone LFOs then for a proper song position sync you will have to use Tempo + Beat and Retrigger set to on. Also the DAW has to be playing.
The H6 preset you mentioned though uses an LFO module. This is a monophonic LFO which won’t produce a second modulation signal when a new note is pressed, so there are never any phase clashes.
If you use the LFO module, the DAW doesn’t need to playback for sync to work as expected.