I don't think a petition is going to amount to anything sadly - players have been begging Soken to loop dungeon and field BGM since the Beta, and he just constantly replies that it's 'working as intended', in that he doesn't want the BGM to overpower the scene but to be ambient, more complimentary, hence why the music starts at the opening to a dungeon or when entering a new zone, but then fades into silence.

I don't agree with it myself, as it goes against FF traditions, but that's life I guess.