According to Wikipedia, it only needs to be 80 times as massive:

Jupiter would need to be about 80 times as massive to fuse hydrogen and become a star.[8]

Six billion tons a second ends up as an increase of about seven orders of magnitude lower than jupiters mass in a year, so it’s still several millions of years away from that point, assuming linear growth. I see no reason to assume linear growth, but this is probably still not going to happen in the foreseeable future.